Abstract
In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field, we found that the moments (mean, variance, and higher moments) of novel statistics on set partitions of [n]={1,2,⋯,n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.
Background
The set partitions of [ n]={1,2,⋯,n} (denoted Π(n)) are a classical object of combinatorics. In studying the character theory of upper triangular matrices (see section ‘Set partitions, enumerative group theory, and super characters’ for background) we were led to some unusual statistics on set partitions. For a set partition λ of n, consider the dimension exponent (Table 1).
where λ has ℓ blocks, M_{i} and m_{i} are the largest and smallest elements of the ith block. How does d(λ) vary with λ? As shown below, its mean and second moment are determined in terms of the Bell numbers B_{n}
Table 1. A table of the dimension exponent f (n ,0, d )
The right hand sides of these formulae are linear combinations of Bell numbers with polynomial coefficients. Dividing by B_{n} and using asymptotics for Bell numbers (see section ‘Asymptotic analysis’) in terms of α_{n}, the positive real solution of ue^{u}=n+1 (so α_{n}= log(n)− log log(n)+⋯) gives
This paper gives a large family of statistics that admit similar formulae for all moments. These include classical statistics such as the number of blocks and number of blocks of size i. It also includes many novel statistics such as d(λ) and c_{k}(λ), the number of k crossings. The number of two crossings appears as the intertwining exponent of super characters.
Careful definitions and statements of our main results are in section ‘Statement of the main results’. Section ‘Set partitions, enumerative group theory, and super characters’ reviews the enumerative and probabilistic theory of set partitions, finite groups, and super characters. Section ‘Computational results’ gives computational results; determining the coefficients in shifted Bell expressions involves summing over all set partitions for small n. For some statistics, a fast new algorithm speeds things up. Proofs of the main theorems are in sections ‘Proofs of recursions, asymptotics, and Theorem 3’ and ‘Proofs of Theorems 1 and 2’. Section ‘More data’ gives a collection of examples  moments of order up to six for d(λ) and further numerical data. In a companion paper [1], the asymptotic limiting normality of d(λ), c_{2}(λ), and some other statistics is shown.
Statement of the main results
Let Π(n) be the set partitions of [n]={1,2,⋯,n} (so Π(n)=B_{n}, the nth Bell number). A variety of codings are described in section ‘Set partitions, enumerative group theory, and super characters’. In this section, λ∈Π(n) is described as λ=B_{1}B_{2}⋯B_{ℓ} with B_{i}∩B_{j}=∅, . Write i∼_{λ}j if i and j are in the same block of λ. It is notationally convenient to think of each block as being ordered. Let First (λ) be the set of elements of [n] which appear first in their block and Last(λ) be the set of elements of [n] which occur last in their block. Finally, let Arc(λ) be the set of distinct pairs of integers (i,j) which occur in the same block of λ such that j is the smallest element of the block greater than i. As usual, λ may be pictured as a graph with vertex set [n] and edge set Arc(λ).
For example, the partition λ=1356274, represented in Figure 1, has First(λ)={1,2,4}, Last(λ)={6,7,4}, and Arc(λ)={(1,3),(3,5),(5,6),(2,7)}.
Figure 1. An example partition λ =1356274.
A statistic on λ is defined by counting the number of occurrences of patterns. This requires some notation.
Definition1
(i) A pattern of length k is defined by a set partition P of [k] and subsets , and . Let .
(ii) An occurrence of a pattern of length k in λ∈Π(n) is s=(x_{1},⋯,x_{k}) with x_{i}∈[ n] such that
1. x_{1}<x_{2}<⋯<x_{k}.
2. x_{i}∼_{λ}x_{j} if and only if i∼_{P}j.
Write if s is an occurrence of in λ.
(iii) A simple statistic is defined by a pattern of length k and . If λ∈Π(n) and , write . Let
Let the degree of a simple statistic be the sum of the length of and the degree of Q.
(iv) A statistic is a finitelinear combination of simple statistics. The degree of a statistic is defined to be the minimum over such representations of the maximum degree of any appearing simple statistic.
Remark1
In the notation above, is the set of firsts elements, is the set of lasts, A is the arc set of the pattern, and is the set of consecutive elements.
Examples
1. Number of blocks in λ:
Here, is a pattern of length 1, , , and Q(y,m)=1. Similarly, the nth moment of ℓ(λ) can be computed using
where is the pattern of length k corresponding to P, the partition of [k] into blocks of size 1, with , and .
2. Number of blocks of size i: define a pattern of length i by: (1) all elements of [i] are equivalent, (2) , (3) , (4) , and (5) . Then,
is the number of i blocks in λ (if i=1, ). Similarly, the moments of the number of blocks of size i is a statistic. See Theorem 1.
3. k crossings: a k crossing [2] of a λ∈Π(n) is a sequence of arcs (i_{t},j_{t})_{1≤t≤k}∈Arc(λ) with
The statistic cr_{k}(λ) which counts the number of k crossings of λ can be represented by a pattern of length 2k with (1) i∼_{P}k+i for i=1,⋯,k, (2) , (3) , and (4) .
Partitions with cr_{2}(λ)=0 are in bijection with Dyck paths and so are counted by the Catalan numbers (see Stanley’s second volume on enumerative combinatorics [3]). Partitions without crossings have proved themselves to be very interesting.
Crossing seems to have been introduced by Krewaras [4]. See Simion’s [5] for an extensive survey and Chen et al. [2] and Marberg [6] for more recent appearances of this statistic. The statistic cr_{2}(λ) appears as the intersection exponent in section ‘Super character theory’.
4. Dimension exponent: the dimension exponent described in the introduction is a linear combination of the number of blocks (a simple statistic of degree 1), the last elements of the blocks (a simple statistic of degree 2), and the first elements of the blocks (a simple statistic of degree 2). Precisely, define where is the pattern of length 1, with , , and Q(y,m)=y. Similarly, let where is the pattern of length 1, with , , and Q(y,m)=y. Then,
5. Levels: the number of levels in λ, denoted f_{levels}(λ), (see page 383 of [7] or Shattuck [8]) is the number of i such that i and i+1 appear in the same block of λ. We have
where is a pattern of length 2 with , , and Q=1.
6. The maximum block size of a partition is not a statistic in this notation.
The set of all statistics on is a filtered algebra.
Theorem1.
Let be the set of all set partition statistics thought of as functions . Then is closed under the operations of pointwise scaling, addition and multiplication. In particular, if and , then there exist partition statistics g_{a},g_{+},g_{∗} so that for all set partitions λ,
Furthermore, deg(g_{a})≤ deg(f_{1}), deg(g_{+})≤ max(deg(f_{1}), deg(f_{2})), and deg(g_{∗})≤ deg(f_{1})+ deg(f_{2}). In particular, is a filteredalgebra under these operations.
Remark2.
Properties of this algebra remain to be discovered.
Definition2.
A shifted Bell polynomial is any function that is zero or can be expressed in the form
where and each such that Q_{I}(x)≠0 and Q_{K}(x)≠0. i.e., it is a finite sum of polynomials multiplied by shifted Bell numbers. Call K the upper shift degree of R and I the lower shift degree of R.
Remark3.
The representation of a shifted Bell polynomial is unique. This can be understood by considering the asymptotics of each individual term as n→∞.
Our first main theorem shows that the aggregate of a statistic is a shifted Bell polynomial.
Theorem2.
For any statistic, f of degree N, there exists a shifted Bell polynomial R such that for all n≥1
Moreover,
1. the upper shift index of R is at most N and the lower shift index is bounded below by −k, where k is the length of the pattern associated f.
2. the degree of the polynomial coefficient of B_{n+N−j} in R is bounded by j for j≤N and by j−1 for j>N.
The following collects the shifted Bell polynomials for the aggregates of the statistics given previously.
Examples
1. Number of blocks in λ:
This is elementary and is established in Proposition 1
2. Number of blocks of size i:
This is also elementary and is established in Proposition 1.
3. Two crossings: Kasraoui [9] established
4. Dimension exponent:
This is given in Theorem 1.
5. Levels: Shattuck [8] showed that
It is amusing that this implies that B_{3n}≡B_{3n+1}≡1 (mod 2) and B_{3n+2}≡0 (mod 2) for all n≥0.
Remark4.
Chapter 8 of Mansour’s book [7] and the research papers [911] contain many other examples of statistics which have shifted Bell polynomial aggregates. We believe that each of these statistics is covered by our class of statistics.
Set partitions, enumerative group theory, and super characters
This section presents background and a literature review of set partitions, probabilistic and enumerative group theory, and super character theory for the upper triangular group over a finite field. Some sharpenings of our general theory are given.
Set partitions
Let Π(n,k) denote the set partitions of n labeled objects with k blocks and Π(n)=∪_{k}Π(n,k); so Π(n,k)=S(n,k) is the Stirling number of the second kind and Π(n)=B_{n} is the nth Bell number. The enumerative theory and applications of these basic objects is developed in the studies of Graham et al. [12], Knuth [13], Mansour [7], and Stanley [14]. There are many familiar equivalent codings.
● Equivalence relations on n objects
● Binary, strictly upper triangular zeroone matrices with no two ones in the same row or column (equivalently, rook placements on a triangular Ferris board (Riordan [15]).
● Arcs on n points
● Restricted growth sequences a_{1},a_{2},…,a_{n}; a_{1}=0,a_{j+1}≤1+ max(a_{1},…,a_{j}) for 1≤j<n (Knuth [13], p. 416)
● Semilabeled trees on n+1 vertices
● Vacillating tableau: a sequence of partitions λ^{0},λ^{1},⋯,λ^{2n} with λ^{0}=λ^{2n}=∅ and λ^{2i+1} is obtained from λ^{2i} by doing nothing or deleting a square and λ^{2i} is obtained from λ^{2i−1} by doing nothing or adding a square (see [2]).
The enumerative theory of set partitions begins with Bell polynomials. Let with X_{i}(λ) the number of blocks in λ of size i; so set and A classical version of the exponential formula gives
These elegant formulae have been used by physicists and chemists to understand fragmentation processes ([16] for extensive references). They also underlie the theory of polynomials of binomial type [17,18], that is, families P_{n}(x) of polynomials satisfying
These unify many combinatorial identities, going back to Faa de Bruno’s formula for the Taylor series of the composition of two power series.
There is a healthy algebraic theory of set partitions. The partition algebra of [19] is based on a natural product on Π(n) which first arose in diagonalizing the transfer matrix for the Potts model of statistical physics. The set of all set partitions has a Hopf algebra structure which is a general object of study in [20].
Crossings and nestings of set partitions is an emerging topic, see [2,21,22] and their references. Given λ∈Π(n) two arcs (i_{1},j_{1}) and (i_{2},j_{2}) are said to cross if i_{1}<i_{2}<j_{1}<j_{2} and nest if i_{1}<i_{2}<j_{2}<j_{1}. Let cr(λ) and ne(λ) be the number of crossings and nestings. One striking result: the crossings and nestings are equidistributed ([21] Corollary 1.5), they show
As explained in section ‘Super character theory’, crossings arise in a group theoretic context and are covered by our main theorem. Nestings are also a statistic.
This crossing and nesting literature develops a parallel theory for crossings and nestings of perfect matchings (set partitions with all blocks of size 2). Preliminary works suggest that our main theorem carry over to matchings with B_{n} reduced to (2n)!/2^{n}n!.
Turn next to the probabilistic side: what does a ‘typical’ set partition ‘look like’? For example, under the uniform distribution on Π(n)
● What is the expected number of blocks?
● How many singletons (or blocks of size i) are there?
● What is the size of the largest block?
The Bell polynomials can be used to get moments. For example:
Proposition1.
(i) Let ℓ(λ) be the number of blocks. Then
(ii) Let X_{1}(λ) be the number of singleton blocks, then
In accordance with our general theorem, the right hand sides of (i) and (ii) are shifted Bell polynomials. To make contact with the results shown previously, there is a direct proof of these classical formulae.
Proof.
Specializing the variables in the generating function (2) gives a two variable generating functions for ℓ:
Differentiating with respect to y and setting y=1 shows that m(ℓ;n) is the coefficient of in . Noting that
yields m(ℓ)=B_{n+1}−B_{n}. Repeated differentiation gives the higher moments.
For X_{1}, specializing variables gives
Differentiation with respect to y and settings y=1 readily yields the claimed results.
The moment method may be used to derive limit theorems. An easier, more systematic method is due to Fristedt [23]. He interprets the factorization of the generating function B(t) in (2) as a conditional independence result and uses ‘dePoissonization’ to get results for finite n. Let X_{i}(λ) be the number of blocks of size i. Roughly, his results say that are asymptotically independent and of size (log(n))^{i}/i!. More precisely, let α_{n} satisfy (so α_{n}= log(n)− log log(n)+o(1)). Let then
where Fristedt also has a description of the joint distribution of the largest blocks.
Remark5.
It is typical to expand the asymptotics in terms of u_{n} where . In this notation, u_{n} and α_{n} differ by O(1/n).
The number of blocks ℓ(λ) is asymptotically normal when standardized by its mean and variance . These are precisely given by Proposition 1. Refining this, Hwang [24] shows
Stam [25] has introduced a clever algorithm for random uniform sampling of set partitions in Π(n). He uses this to show that if W(i) is the size of the block containing i, 1≤i≤k, then for k finite and n large W(i) are asymptotically independent and normal with mean and variance asymptotic to α_{n}. In [1], we use Stam’s algorithm to prove the asymptotic normality of d(λ) and cr_{2}(λ).
Any of the codings previously mentioned lead to distribution questions. The upper triangular representation leads to the study of the dimension and crossing statistics, the arc representation suggests crossings, nestings, and even the number of arcs, i.e. n−ℓ(λ). Restricted growth sequences suggest the number of zeros, the number of leading zeros, largest entry. See Mansour [7] for this and much more. Semilabeled trees suggest the number of leaves, length of the longest path from root to leaf, and various measures of tree shape (e.g., max degree). Further probabilistic aspects of uniform set partitions can be found in [16,26].
Probabilistic group theory
One way to study a finite group G is to ask what ‘typical’ elements ‘look like’. This program was actively begun by Erdös and Turan [2733] who focused on the symmetric group S_{n}. Pick a permutation σ of n at random and ask the following:
● How many cycles in σ? (about logn)
● What is the length of the longest cycle? (about 0.61n)
● How many fixed points in σ? (about 1)
● What is the order of σ? (roughly )
In these and many other cases, the questions are answered with much more precise limit theorems. A variety of other classes of groups have been studied. For finite groups of Lie type, see [34] for a survey and [35] for wideranging applications. For p groups, see [36].
One can also ask questions about ‘typical’ representations. For example, fix a conjugacy class C (e.g., transpositions in the symmetric group), what is the distribution of χ_{ρ}(C) as ρ ranges over irreducible representations [34,37,38]. Here, two probability distributions are natural, the uniform distribution on ρ and the Plancherel measure ( with d_{ρ} the dimension of ρ). Indeed, the behavior of the ‘shape’ of a random partition of n under the Plancherel measure for S_{n} is one of the most celebrated results in modern combinatorics. See Stanley’s study [39] for a survey with references to the work of Kerov and Vershik [40], Logan and Shepp [41], Baik et al. [42], and many others.
The previous discussion focuses on finite groups. The questions make sense for compact groups. For example, pick a random matrix from Haar measure on the unitary group U_{n} and ask: what is the distribution of its eigenvalues? This leads to the very active subject of random matrix theory. We point to the wonderful monographs of Anderson et al. [43], and Forrester [44] which have extensive surveys.
Super character theory
Let G_{n}(q) be the group of n×n matrices which are upper triangular with ones on the diagonal over the field . The group G_{n}(q) is the Sylow p subgroup of for q=p^{a}. Describing the irreducible characters of G_{n}(q) is a wellknown wild problem. However, certain unions of conjugacy classes, called superclasses, and certain characters, called supercharacters, have an elegant theory. In fact, the theory is rich enough to provide enough understanding of the Fourier analysis on the group to solve certain problems, see the work of AriasCastro et al. [45]. These superclasses and supercharacters were developed by André [4648] and Yan [49]. Supercharacter theory is a growing subject. See [6,5054] and their references.
For the groups G_{n}(q), the supercharacters are determined by a set partition of [ n] and a map from the set partition to the group . In the analysis of these characters, there are two important statistics, each of which only depends on the set partition. The dimension exponent is denoted d(λ), and the intertwining exponent is denoted i(λ).
Indeed, if χ_{λ} and χ_{μ} are two supercharacters, then
While d(λ) and i(λ) were originally defined in terms of the upper triangular representation (for example, d(λ) is the sum of the horizontal distance from the ‘ones’ to the super diagonal), their definitions can be given in terms of blocks or arcs:
and
Remark6.
Notice that i(λ)=cr_{2}(λ) is the number of two crossings which were introduced in the previous sections.
Our main theorem shows that there are explicit formulae for every moment of these statistics. The following represents a sharpening using special properties of the dimension exponent.
Theorem3.
For each k∈{0,1,2,⋯ }, there exists a closed form expression
where each P_{k,2k−j} is a polynomial with rational coefficients. Moreover, the degree of P_{k,2k−j} is
For example,
Remark7.
See section ‘More data’ for the moments with k≤6, and see [55] for the moments with k≤22. The first moment may be deduced easily from results of Bergeron and Thiem [56]. Note that they seem to have an index which differs by one from ours.
Remark8.
Theorem 3 is stronger than what is obtained directly from Theorem 2. For example, the lower shift index is 0, while the best that can be obtained from Theorem 2 is a lower shift index of −k. This theorem is proved by working directly with the generating function for a generalized statistic on ‘marked set partitions’. These set partitions are introduced in section ‘Computational results’.
Asymptotics for the Bell numbers yield the following asymptotics for the moments. The following result gives some asymptotic information about these moments.
Theorem4.
Let α_{n}= log(n)− log log(n)+o(1)be the positive real solution of ue^{u}=n+1. Then
Let be the symmetrized moments of the dimension exponent. Then
Remark9.
Asymptotics for S_{k}(d;n) with k=1,2,3,4,5,6 and with further accuracy are in section ‘More data’.
Analogous to these results for the dimension exponent are the following results for the intertwining exponent.
Theorem5.
For each k∈{0,1,2,⋯ } there exists a closed form expression
where each Q_{k,2k−j} is a polynomial with rational coefficients. Moreover, the degree of Q_{k,2k−j} is bounded by j.
For example,
Remark10.
The expression for M(i;n)=M(cr_{2};n) was established first by Kasraoui (Theorem 2.3 of [9]).
Remark11.
Theorem 5 is deduced directly from Theorem 2. The shifted Bell polynomials for M(i^{k};n) for k≤5 are given in section ‘More data’, and see [55] for the aggregates with k≤12.
Remark12.
Amusingly, the formula for M(i;n) implies that the sequence taken modulo 4 is periodic of length 12 beginning with {1,1,2,1,3,0,3,1,0,3,3,2}. Similarly, the formula for M(i^{2};n) shows that the sequence is periodic modulo 9 (respectively 16) with period 39 (respectively 48). For more about such periodicity, see the papers of Lunnon et al. [57] and Montgomery et al. [58].
In analogy with Theorem 4, there is the following asymptotic result.
Theorem6.
With α_{n} as above,
Theorems 3 and 5 show that there will be closed formulae for all of the moments of these statistics. Moreover, these theorems give bounds for the number of terms in the summand and the degree of each of the polynomials. Therefore, to compute the formulae, it is enough to compute enough values for M(d^{k};n) or M(i^{k};n) and then to do linear algebra to solve for the coefficients of the polynomials. For example, M(d;n) needs P_{1,2}(n) which has degree at most 0, P_{1,1}(n) which has degree at most 1, and P_{1,0}(n) which has degree at most 0. Hence, there are four unknowns, and so only M(d;n) for n=1,2,3,4 are needed to derive the formula for the expected value of the dimension exponent.
Computational results
Enumerating set partitions and calculating these statistics would take time O(B_{n}) (see Knuth’s volume [13] for discussion of how to generate all set partitions of fixed size, the book of Wilf and Nijenhuis [59], or the website [60] of Ruskey). This section introduces a recursion for computing the number of set partitions of n with a given dimension or intertwining exponent in time O(n^{4}). The recursion follows by introducing a notion of ‘marked’ set partitions. This generalization seems useful in general when computing statistics which depend on the internal structure of a set partition.
The results may then be used with Theorems 3 and 5 to find exact formulae for the moments. Proofs are given in section ‘Proofs of recursions, asymptotics, and Theorem 3’.
For a set partition λ, mark each block either open or closed. Call such a partition a marked set partition. For each marked set partition λ of [ n], let o(λ) be the number of open blocks of λ and ℓ(λ) be the total number of blocks of λ. (Marked set partitions may be thought of as what is obtained when considering a set partition of a potentially larger set and restricting it to [n]. The open blocks are those that will become larger upon adding more elements of this larger set, while the closed blocks are those that will not.) With this notation, define the dimension of λ with blocks B_{1},B_{2},⋯ by
It is clear that if o(λ)=0, then λ may be thought of as a usual ‘unmarked’ set partition and is the dimension exponent of λ. Define
Theorem7.
For n>0
with initial condition f(0;A,B)=0 for all (A,B)≠(0,0) and f(0;0,0)=1.
Therefore, to find the number of partitions of [ n] with dimension exponent equal to k, it suffices to compute f(n,0,k) for k and n. Figure 2 gives the histograms of the dimension exponent when n=20 and n=100. With increasing n, these distributions tend to normal with mean and variance given in Theorem 4. This approximation is already apparent for n=20.
Figure 2. Histograms of the dimension exponent counts for n =20 and n =100.
It is not necessary to compute the entire distribution of the dimension index to compute the moment formulae for the dimension exponent. Namely, it is better to implement the following recursion for the moments.
Corollary1.
To compute M(d^{k};n), then for each m<n, this recursion allows us to keep only k values rather than computing all O(m·m^{2}) values of f(m,A,B). To find the linear relation of Theorem 3, only O(k·k^{2}) values of M_{k}(d;n,A) are needed.
In analogy, there is a recursion for the intertwining exponent.
Let f_{(i)}(n,A,B) be the number of marked partitions of [ n] with intertwining weight equal to B and with A open sets where the intertwining weight is equal to the number of interlaced pairs i⌢j and k⌢ℓ where k is in a closed set plus the number of triples i,k,j such that i⌢j and k is in an open set.
Theorem8.
With the notation above, the following recursion holds
This recursion allows the distribution to be computed rapidly. Figure 3 gives the histograms of the intertwining exponent when n=20 and n=100. Again, for increasing n, the distribution tends to normal with mean and variance from Theorem 6. The skewness is apparent for n=20.
Figure 3. Histograms of the intertwining exponent counts for n =20 and n =100.
Proofs of recursions, asymptotics, and Theorem 3
This section gives the proofs of the recursive formulae discussed in Theorems 7 and 8 Additionally, this section gives a proof of Theorem 3 using the threevariable generating function for f(n,A,B). Finally, it gives an asymptotic expansion for B_{n+k}/B_{n} with k fixed and n→∞. This asymptotic is used to deduce Theorems 4 and 6.
Recursive formulae
This subsection gives the proof of the recursions for f(n,A,B) and f_{(i)}(n,A,B) given in Theorems 7 and 8 The recursion is used in the next subsection to study the generating function for the dimension exponent.
Proof of Theorem 7 The four terms of the recursion come from considering the following cases: (1) n is added to a marked partition of [ n−1] as a singleton open set, (2) n is added to a marked partition of [ n−1] as a singleton closed set, (3) n is added to an open set of a marked partition of [ n−1] and that set remains open, and (4) n is added to an open set of a marked partition of [ n−1] and that set is closed.
Proof of Theorem 8 The argument is similar to that of Theorem 7. The same four cases arise. However, when adding n to an open set, the statistic may increase by any value j and it does so in exactly one way.
The generating function for f(n,A,B)
This section studies the generating function for f(n,A,B) and deduces Theorem 3. Let
be the threevariable generating function. Theorem 7 implies that
Then, F(X,0,Z) is the generating function for the distribution of d(λ), i.e.,
Thus, the generating function for the kth moment is
Consider
Lemma1
In the notation above,
Proof.
From (8),
Hence, solving for F_{n} gives
Throughout the remainder Y=e^{α}−1. Abusing notation, let
The following lemma gives an expression for G_{k}(X,α) in terms of a differential operators. Define the operators
Lemma2.
Clearly, G_{0}(X,Y)=1. Moreover,
Proof.
(10) is equivalent to
Now
where a e^{α} has been commuted through. Then,
Since G_{k}(0,α)=0 for k>0,
From this
for some constants
The next lemma evaluates the terms in the summation of Lemma 2, thus yielding a generating function for G_{k}(X,Y) which resembles that for the Bell numbers.
Lemma3.
Proof.
It is easy to see by induction on ℓ that T^{ℓ}1 is a polynomial in e^{X+α}. Thus,
Hence
From this, it is easy to see that
And the result follows.
Lemmas 2 and 3 readily yield the following expression for the moments of the dimension exponent as a shifted Bell polynomial.
Lemma4.
For each k≥0 and n≥0
Theorem 3 needs some further constraints on the degrees of terms in this polynomial. The following lemma yields the claimed bounds for the degrees.
Lemma5.
In the notation above, unless all of the following hold:
1. c≤b.
2. c<b unless a=0.
3. b≤2k.
4. 3c−b≤k.
5. 3c−b≤k−2 if a≠0.
Proof.
Let H_{a,b,c}(X,α)=S^{a}T^{b}X^{c}1. Using Equation 12, write in terms of the for ℓ<k. To do this requires understanding
As a first claim: if a=0, then the above is simply . This is seen easily from the fact that R commutes with T. For a≠0, it is easy to see that this is a linear combination of the over c^{′}≤c, and of over b^{′}≤b.
The desired properties can now be proved by induction on k. It is clear that they all hold for k=0. For larger k, assume that they hold for all k−ℓ, and use Equation 12 to prove them for k.
By the inductive hypothesis, the TG_{k−ℓ} are linear combinations of H_{a,b,c} with c<b. Thus, (T−TS^{−1}−S)^{ℓ}TG_{k−ℓ} is a linear combination of H_{a,b,c}’s with b>c. Thus, by Equation 12, G_{k} is a linear combination of H_{a,b,c}’s with c≤b and a=0 or with c<b. This proves properties 1 and 2.
By the inductive hypothesis, the G_{k−ℓ} are linear combinations of H_{a,b,c} with b≤2(k−ℓ). Thus, T−TS^{−1}−S^{ℓ}TG_{k−ℓ} is a linear combination of H_{a,b,c}’s with b≤2k+1−ℓ≤2k. Thus, by Equation 12, G_{k} is a linear combination of H_{a,b,c}’s with b≤2k. This proves property 3.
Finally, consider the contribution to G_{k} coming from each of the G_{k−ℓ} terms. For ℓ=1, G_{k−ℓ} is a linear combination of H_{a,b,c}’s with 3c−b≤k−3 if a≠0, 3c−b≤k−1 if a=0. Thus, TG_{k−ℓ} is a linear combination of H_{a,b,c}’s with 3c−b≤k−3 if a≠0, and 3c−b≤k−2 otherwise. Thus, (T−TS^{−1}−S)^{ℓ}TG_{k−ℓ} is a linear combination of H_{a,b,c}’s with 3c−b≤k−3 if a=0, and 3c−b≤k−2 otherwise. Thus, the contribution from these terms to G_{k} is a linear combination of H_{a,b,c}’s with 3c−b≤k and 3c−b≤k−2 if a≠0. For the terms with ℓ>1, G_{k−ℓ} is a linear combination of H_{a,b,c}’s with 3c−b≤k−2 and 3c−b≤k−4 when a≠0. Thus, TG_{k−ℓ} is a linear combination of H_{a,b,c}’s with 3c−b≤k−3, as is (T−TS^{−1}−S)^{ℓ}TG_{k−ℓ}. Thus, the contribution of these terms to G_{k} is a linear combination of H_{a,b,c}’s with 3c−b≤k and 3c−b≤k−3 if a≠0. This proves properties 4 and 5.
This completes the induction and proves the Lemma.
From this Lemma, it is easy to see that
for some polynomials P_{k,ℓ}(n) with deg(P_{k,ℓ})≤ min(2k−ℓ,k/2+ℓ/2).
Asymptotic analysis
This section presents some asymptotic analysis of the Bell numbers and ratios of Bell numbers. These results yield Theorems 4 and 6. Similar analysis can be found in [13].
Proposition2.
Let α_{n} be the solution to
and let
Then
More precisely, for T≥0
where R_{m,k} are rational functions. In particular
Proof.
The proof is very similar to the traditional saddle point method for approximating B_{n}. The idea is to evaluate at the saddle point for B_{n} rather than for B_{n+k}. We follow the proof in Chapter 6 of [61].
By Cauchy’s formula,
where C encircles the origin once in the positive direction. Deform the path to a vertical line u−i∞ to u+i∞ by taking a large segment of this line and a large semicircle going around the origin. As the radius, say R, is taken to infinity the factor z^{−n−k−1}=O(R^{−n−k−1}) and exp(e^{z}) is bounded in the halfplane.
Choose u=α_{n} and then
where
The real part has maxima around y=2πm for each integer m, but using for π<y<α_{n} and for y>α_{n} as in [61] gives
Next, note that
Making the change of variables and extending the sum of interval of integration gives
Hence, Taylor expanding around y=0 and using
gives the desired result.
For more details, see [61].
Proposition 2 yields
Direct application of this result gives the results in Theorems 4 and 6.
Proofs of Theorems 1 and 2
This section gives the proofs of Theorems 2 and 1. Theorem 2 implies Theorem 5. A pair of lemmas which will be useful in the proof of Theorem 2:
Lemma6.
For B_{n}, the Bell numbers, define
where r,d,k,s are nonnegative integers. Then, g_{r,d,k,s}(n) is a shifted Bell polynomial of lower shift index −k and upper shift index r+s−k.
Proof.
It clearly suffices to prove that g_{r,0,k,s}(n) is a shifted Bell polynomial. Since g_{r,0,0,s}(n−k)=g_{r,0,k,s}(n), it suffices to prove that g_{r,s}(n):=g_{r,0,0,s}(n) is a shifted Bell polynomial.
For this, consider the exponential generating function
This is easily seen to be equal to times a polynomial in e^{x} of degree s+r.
Since g_{0,s}(n)=B_{n+s}, the generating function equals times a polynomial of exact degree s. From this, for all s,r the space of all polynomials in e^{x} of degree at most s+r times is spanned by the set of generating functions as m runs over all integers 0,1,…,s+r. Since the generating function for g_{r,s}(n) lies in this span,
for some rational numbers β_{s,r,m}. It follows that for all n,
For a sequence, r={r_{0},r_{1},⋯,r_{k}}, of rational numbers and a polynomial define
where x_{0}=0,x_{k+1}=n+1.
Lemma7.
Fix k, let and r={r_{0},r_{1},⋯,r_{k}} be a sequence of rational numbers. As defined above, M(k,Q,r,n,x) is a rational linear combination of terms of the form
Proof.
The proof is by induction on k. If k=0 then definitionally, M(k,Q,r,n,x)=Q(n)(x+r_{0})^{n}, providing a base case for our result. Assume that the lemma holds for k one smaller. For this, fix the values of x_{1},…,x_{k−1} in the sum and consider the resulting sum over x_{k}. Then
Consider the inner sum over x_{k}:
If r_{k−1}=r_{k}, then the product of the last two terms is always , and thus the sum is some polynomial in x_{1},…,x_{k−1},n times . The remaining sum over x_{1},…,x_{k−1} is exactly of the form M(k−1,Q^{′},r^{′},n−1,x), for some polynomial Q^{′}, and thus, by the inductive hypothesis, of the correct form.
If r_{k−1}≠r_{k}, the sum is over pairs of nonnegative integers a=x_{k}−x_{k−1}−1 and b=n−x_{k}−1 summing to n−x_{k−1}−2 of some polynomial, Q^{′} in a and n and the other x_{i} times (x+r_{k−1})^{a}(x+r_{k})^{b}. Letting y=(x+r_{k−1}) and z=(x+r_{k}), this is a sum of Q^{′}(x_{i},n,a)y^{a}z^{b}. Let d be the a degree of Q^{′}. Multiplying this sum by (y−z)^{d+1}, yields, by standard results, a polynomial in y and z of degree n−x_{k−1}−2+(d+1) in which all terms have either y exponent or z exponent at least n−x_{k−1}−1. Thus, this inner sum over x_{k} when multiplied by the nonzero constant (r_{k−1}−r_{k})^{d+1} yields the sum of a polynomial in x,n,x_{1},…,x_{k−1} times plus another such polynomial times . Thus, M(k,Q,r,n,x) can be written as a linear combination of terms of the form G(x)M(k−1,Q^{′},r^{′},n,x). The inductive hypothesis is now enough to complete the proof.
Turn next to the proof of Theorem 2. Proof of Theorem 2
It suffices to prove this Theorem for simple statistics. Thus, it suffices to prove that for any pattern P and polynomial Q that
is given by a shifted Bell polynomial in n. As a first step, interchange the order of summation over s and λ above. Hence,
To deal with the sum over λ above, first consider only the blocks of λ that contain some element of s. Equivalently, let λ^{′} be obtained from λ by replacing all of the blocks of λ that are disjoint from s by their union. To clarify this notation, let Π^{′}(n) denote the set of all set partitions of [ n] with at most one marked block. For λ^{′}∈Π^{′}(n), say that s∈_{P}λ^{′} if s in an occurrence of P in λ^{′} as a regular set partition so that additionally the nonmarked blocks of λ^{′} are exactly the blocks of λ^{′} that contain some element of s. For λ^{′}∈Π^{′}(n) and λ∈Π(n), say that λ is a refinement of λ^{′} if the unmarked blocks in λ^{′} are all parts in λ, or equivalently, if λ can be obtained from λ^{′} by further partitioning the marked block. Denote λ being a refinement of λ^{′} as λ⊩λ^{′}. Thus, in the previous computation of M(f_{P,Q};n), letting λ^{′} be the marked partition obtained by replacing the blocks in λ disjoint from s by their union:
Note that the λ in the final sum above correspond exactly to the set partitions of the marked block of λ^{′}. For λ^{′}∈Π^{′}(n), let λ^{′} be the size of the marked block of λ^{′}. Thus,
Remark13.
This is valid even when the marked block is empty.
Dealing directly with the Bell numbers above will prove challenging, so instead compute the generating function
After computing this, extract the coefficients of M(P,Q,n,x) and multiply them by the appropriate Bell numbers.
To compute M(P,Q,n,x), begin by computing the value of the inner sum in terms of s=(x_{1}<x_{2}<…<x_{k}) that preserves the consecutivity relations of P (namely those in C(P)). Denote the equivalence classes in P by 1,2,…,ℓ. Let z_{i} be a representative of this ith equivalence class. Then, an element λ^{′}∈Π^{′}(n) so that s∈_{P}λ^{′} can be thought of as a set partition of [ n] into labeled equivalence classes 0,1,…,ℓ, where the 0th class is the marked block, and the ith class is the block containing . Thus, think of the set of such λ^{′} as the set of maps g:[ n]→{0,1,…,ℓ} so that:
1. g(x_{j})=i if j is in the ith equivalence class
2. g(x)≠i if x<x_{j}, j∈F(P) and j is in the ith equivalence class
3. g(x)≠i if x>x_{j}, j∈L(P) and j is in the ith equivalence class
4. g(x)≠i if , (j,j^{′})∈A(P) and j,j^{′} are in the ith equivalence class
It is possible that no such g will exist if one of the latter three properties must be violated by some x=x_{h}. If this is the case, this is a property of the pattern P, and not the occurrence s, and thus, M(f_{P,Q};n)=0 for all n. Otherwise, in order to specify g, assign the given values to g(x_{i}) and each other g(x) may be independently assigned values from the set of possibilities that does not violate any of the other properties. It should be noted that 0 is always in this set, and that furthermore, this set depends only which of the x_{i} our given x is between. Thus, there are some sets S_{0},S_{1},…,S_{k}⊆{0,1,…,ℓ}, depending only on s, so that g is determined by picking functions
Thus, the sum over such λ^{′} of is easily seen to be
where r_{i}=S_{i}−1 (recall S_{i}>0, because 0∈S_{i}). For such a sequence, r of rational numbers define
where, as in Lemma 7, using the notation x_{0}=0,x_{k+1}=n+1.
Note that the sum is empty if C(P) contains nonconsecutive elements. We will henceforth assume that this is not the case. We call j a follower if either (j−1,j) or (j,j−1) are in C(P). Clearly, the values of all x_{i} are determined only by those x_{i} where i is not a follower. Furthermore, Q is a polynomial in these values and n. If j is the index of the ith nonfollower then let y_{i}=x_{j}−j+i. Now, sequences of x_{i} satisfying the necessary conditions correspond exactly to those sequences with 1≤y_{1}<y_{2}<⋯<y_{k−f}≤n−f where f is the total number of followers. Thus,
where the are modified versions of the r_{i} to account for the change from {x_{j}} to {y_{i}}. In particular, if x_{j} is the (i+1)^{st} nonfollower, then .
By Lemma 7, is a linear combination of terms of the form F(n)G(x)(x+r_{i})^{n−k} for polynomials and .
Thus, M(f_{P,Q};n) can be written as a linear combination of terms of the form g_{r,d,ℓ,s}(n) where ℓ is the number of equivalence classes in P and r,d,s are nonnegative integers. Therefore, by Lemma 6 M(f_{P,Q};n) is a shifted Bell polynomial.
The bound for the upper shift index follows from the fact that M(f_{P,Q};n)=O(n^{N}B_{n}) and by (13) each term n^{α}B_{n+β} is of an asymptotically distinct size. To complete the proof of the result it is sufficient to bound the lower shift index of the Bell polynomial. By (15) it is clear the largest power of x in each term is (n−k). Thus, from Lemma 6, the resulting shift Bell polynomials can be written with minimum lower shift index −k. This completes the proof.
Next turn to the proof of Theorem 1. To this end, introduce some notation.
Definition3.
Given three patterns P_{1},P_{2},P_{3}, of lengths k_{1},k_{2},k_{3}, say that a merge of P_{1} and P_{2} onto P_{3} is a pair of strictly increasing functions m_{1}:[ k_{1}]→[k_{3}], m_{2}:[ k_{2}]→[ k_{3}] so that
1. m_{1}([k_{1}])∪m_{2}([k_{2}])=[ k_{3}]
2.
if and only if , and if and only if
3. i∈F(P_{3}) if and only if there exists either a j∈F(P_{1}) so that i=m_{1}(j) or a j∈F(P_{2}) so that i=m_{2}(j)
4. i∈L(P_{3}) if and only if there exists either a j∈L(P_{1}) so that i=m_{1}(j) or a j∈L(P_{2}) so that i=m_{2}(j)
5. (i,i^{′})∈A(P_{3}) if and only if there exists either a (j,j^{′})∈A(P_{1}) so that i=m_{1}(j) and i^{′}=m_{1}(j^{′}) or a (j,j^{′})∈A(P_{2}) so that i=m_{2}(j) and i^{′}=m_{2}(j^{′})
6. (i,i^{′})∈C(P_{3}) if and only if there exists either a (j,j^{′})∈C(P_{1}) so that i=m_{1}(j) and i^{′}=m_{1}(j^{′}) or a (j,j^{′})∈C(P_{2}) so that i=m_{2}(j) and i^{′}=m_{2}(j^{′})
Such a merge is denoted as m_{1},m_{2}:P_{1},P_{2}→P_{3}.
Note that the last four properties above imply that given P_{1} and P_{2}, a merge (including a pattern P_{3}) is uniquely defined by maps m_{1},m_{2} and an equivalence relation satisfying (1) and (2) above.
Lemma8.
Let P_{1} and P_{2} be patterns. For any λ there is a onetoone correspondence:
Moreover, under this correspondence
Proof.
Begin by demonstrating the bijection defined by Eq. 16. On the one hand, given given by and m_{1},m_{2}:P_{1},P_{2}→P_{3}, define s_{1} and s_{2} by the sequences and . It is easy to verify that these are occurrences of the patterns P_{1} and P_{2} and furthermore that Eq. 17 holds for this mapping.
This mapping has a unique inverse: Given s_{1} and s_{2}, note that s_{3} must equal the union s_{1}∪s_{2}. Furthermore, the maps m_{a}, for a=1,2, must be given by the unique function so that m_{a}(i)=j if and only if the i^{th} smallest element of s_{a} equals the j^{th} smallest element of s_{3}. Note that the union of these images must be all of [k_{3}]. In order for s_{3} to be an occurrence of P_{3} the equivalence relation must be that if and only if the i^{th} and j^{th} elements of s_{3} are equivalent under λ. Note that since S_{1} and S_{2} were occurrences of P_{1} and P_{2}, that this must satisfy condition (2) for a merge. The rest of the data associated to P_{3} (namely F(P_{3}),L(P_{3}), A(P_{3}), and C(P_{3})) is now uniquely determined by m_{1},m_{2},P_{1},P_{2} and the fact that P_{3} is a merge of P_{1} and P_{2} under these maps. To show that s_{3} is an occurrence of P_{3} first note that by construction the equivalence relations induced by λ and P_{3} agree. If i∈F(P_{3}), then there is a j∈F(P_{a}) with i=m_{a}(j) for some a,j. Since s_{a} is an occurrence of P_{a}, this means that the j^{th} smallest element of s_{a} in in First(λ). On the other hand, by the construction of m_{a}, this element is exactly . This if i∈F(P_{3}), z_{i}∈First(λ). The remaining properties necessary to verify that S_{3} is an occurrence of P_{3} follow similarly. Thus, having shown that the above map has a unique inverse, the proof of the lemma is complete.
Recall, the number of singleton blocks is denoted X_{1} and it is a simple statistic. To illustrate this lemma return to the example of discussed prior to the lemma. Let P_{1}=P_{2} be the pattern of length 1 with A(P_{1})=ϕ, F(P_{1})=L(P_{1})=1. Then there are five possible merges of P_{1} and P_{2} into some pattern P_{3}. The first choice of P_{3} is P_{1} itself. In which case m_{1}(1)=m_{2}(1)=1. The latter choices of P_{3} is the pattern of length 2 with F(P_{3})=L(P_{3})={1,2},A(P_{3})=∅. The equivalence relation on P_{3} could be either the trivial one or the one that relates 1 and 2 (though in the latter case the pattern P_{3} will never have any occurrences in any set partition). In either of these cases, there is a merge with m_{1}(1)=1 and m_{2}(1)=2 and a second merge with m_{1}(1)=2 and m_{2}(1)=1. As a result,
Proof of Theorem 1 The fact that statistics are closed under pointwise addition and scaling follows immediately from the definition. Similarly, the desired degree bounds for these operations also follow easily. Thus only closure and degree bounds for multiplication must be proved. Since every statistic may be written as a linear combination of simple statistics of no greater degree, and since statistics are closed under linear combination, it suffices to prove this theorem for a product of two simple statistics. Thus let f_{i} be the simple statistic defined by a pattern P_{i} of size k_{i} and a polynomial Q_{i}. It must be shown that f_{1}(λ)f_{2}(λ) is given by a statistic of degree at most k_{1}+k_{2}+ deg(Q_{1})+ deg(Q_{2}).
For any λ
Simplify this equation using Lemma 8, writing this as a sum over occurrences of only a single pattern in λ.
Applying Lemma 8,
Thus, the product of f_{1} and f_{2} is a sum of simple characters. Note that the quantity is a polynomial of s_{3} which is denoted Finally, each pattern P_{3} has size at most k_{1}+k_{2} and each polynomial has degree at most deg(Q_{1})+ deg(Q_{2}). Thus the degree of the product is at most the sum of the degrees.
More data
This section contains some data for the dimension and intertwining exponent statistics. The moment formulae of Theorem 3 for k≤22 and the moment formulae for the intertwining exponent for k≤12 have been computed and are available at [55]. Moreover, the values f(n,0,B) for n≤238 and f_{(i)}(n,0,B) for n≤146 are available. These sequences can also be found on Sloane’s Online Encyclopedia of integer sequences [62].
The remainder of this section contains a small amount of data and observations regarding the distributions f(n,0,B) and f_{(i)}(n,0,B) and regarding the shifted Bell polynomials of Theorems 3 and 5.
Dimension index
A couple of easy observations: it is clear that
That is the number of set partitions of [n] with dimension exponent 0 is 2^{n−1}. Set partitions of [ n] that have dimension exponent 0 must have n appearing in a singleton set or it must appear in a set with n−1, thus the result is obtained by recursion. Additionally, the number of set partitions of [ n] with dimension exponent equal to 1 is n2^{n−1}, that is
Curiously, the numbers f(n,0,B) are smooth (roughly they have many small prime factors), for reasonably sized B. This can be established by using the recursion of Theorem 7. For example,
Note that f(100,0,979) has 111 digits and f(100,0,2079) has about 100 digits.
In the notation of Theorem 3, these are some values of the first few moments of d(λ).
These formulae exhibit a number of properties. Here is a list of some of them.
1. Using the fact that B_{n+k}≈n^{k}B_{n}, each moment M(d^{k};n) has a number of terms with asymptotic of size equal to n^{2k}B_{n}, up to powers of log(n) (or α_{n}). Call these terms the leading powers of n. The leading ‘power’ of n contribution is equal to
where T is the operator given by TB_{m}=B_{m+1}. For example, the leading order n contributions for the average is
and the leading order contribution for the second moment is
Structure of this sort is necessary because of the asymptotic normality of the dimension exponent (see the forthcoming work [1]). The next remark also concerns this sort of structure.
2. The next order n terms of M(d^{k};n) have size roughly n^{2k−1}B_{n} and have the shape
where the constants C_{j} are
3. The generating function for the polynomials P_{0,k}(n) seems to be
We do not have a proof of this observation.
As in the introduction, let From Proposition 2 and the formulae for M(d^{k};n) deduced from Theorem 3 and stated in section ‘Dimension index’ and using SAGE, the asymptotic expansion of the first few S_{k} are as follows:
Remark14.
These asymptotics support the claim that the dimension exponent is normally distributed with mean asymptotic to and standard deviation . This result will be established in the forthcoming work [1].
Intertwining index
Table 2 contains the distribution for of the intertwining exponent for the first few n.
Table 2. A table of the distribution of the intertwining exponentf_{(i})(n ,0, B )
In the notation of Theorem 5, these are some values of the first few moments of i(λ)=cr_{2}(λ).
We conjecture that Q_{j}(n)=0 for all j<0.
The formulae stated previously for with (13) give
Acknowledgements
The authors thank the careful referee for a detailed report.
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