Partition Statistics and q-Bell Numbers (q = −1)

Carl G. Wagner Department of Mathematics University of Tennessee Knoxville, TN 37996-1300, USA

                      wagner@math.utk.edu

Abstract

We study three partition statistics and the q-Stirling and q-Bell numbers that serve as their generating functions, evaluating these numbers when q = −1. Among the numbers that arise in this way are (1) Fibonacci numbers and (2) numbers occurring in the study of fermionic oscillators.

1 Introduction

The notational conventions of this paper are as follows: ℕ := {0, 1, 2, … }, ℙ := {1, 2, … },

[0] := ∅, and [n] := {1, …, n} for n ∈ ℙ. Empty sums take the value 0 and empty products the value 1, with 0^{0} := 1. The letter q denotes an indeterminate, with 0_{q} := 0, n_{q} := 1 + q + ⋯ + q^{n−1} for n ∈ ℙ, 0 !_{q} := 1, and n !_{q} := 1_{q}2_{q} ⋯ n_{q} for n ∈ ℙ. The binomial

coefficient \binom{n}{k} is equal to zero if k is a negative integer or if 0 ⩽ n < k.

Let Δ be a finite set of discrete structures, with I : Δ → ℕ. The generating function

G(I, Δ; q) := ∑_{δ∈Δ} q^{I(δ)} = ∑_{k} |{δ ∈ Δ : I(δ) = k}| q^{k}

is a useful tool for studying the statistic I. Elementary examples include the binomial theorem,

(1 + q)^{n} = ∑_{S⊂[n]} q^{|S|} = ∑_{k=0}^{n} \binom{n}{k}q^{k},  (1)

and

n !_{q} = ∑_{σ∈S_{n}} q^{i(σ)},  (2)

where S_{n} is the set of permutations of [n] and i(σ) is the number of inversions in the permutation σ = i_{1}i_{2} … i_{n}, i.e., the number of pairs (r, s) with 1 ⩽ r < s ⩽ n and i_{r} > i_{s} [7, Corollary 1.3.10].

Of course, G(I, Δ; 1) = |Δ|. On the other hand,

G(I, Δ; −1) = |{δ ∈ Δ : I(δ) is even}| − |{δ ∈ Δ : I(δ) is odd}| .  (3)

Hence if G(I, Δ; −1) = 0, the set Δ is “balanced” with respect to the parity of I. In particular, setting q = −1 in (1) yields the familiar result that a finite nonempty set has as many subsets of odd cardinality as it has subsets of even cardinality. Setting q = −1 in (2) reveals that if n ⩾ 2, then among the permutations of [n] there are as many with an odd number of inversions as there are with an even number of inversions.

In this note we consider three q-generalizations of Stirling numbers of the second kind, denoted S_{q}^{∗}(n, k), S_{q}(n, k), and S̃_{q}(n, k). These polynomials are generating functions for three closely related statistics on the set of partitions of [n] with k blocks. Most of the properties of these q-Stirling numbers, to be established below in § 3, have appeared in the literature in various contexts, Carlitz [1] having apparently been the first to construe these numbers as generating functions for partition statistics. See also [3], [4], [8], and [9]. Our aim here is to offer a compact, unified treatment of these numbers. Our analysis is facilitated by a powerful formal algebraic result of Comtet [2].

We derive in § 4 new results on the evaluation of S_{q}^{∗}(n, k), S_{q}(n, k), and S̃_{q}(n, k) and their associated q-Bell numbers (gotten by summing q-Stirling numbers over k for fixed n) when q = −1. Apart from the interpretation of these results in terms of (3), the evaluation of S_{−1}(n, k) and its associated Bell numbers may be of additional interest, since these numbers arise in the study of fermionic oscillators [6].

In § 5 we discuss an alternative approach to establishing our results by means of bijective proofs.

In § 6 the numbers S_{−1}^{∗}(n, k), S_{−1}(n, k), and S̃_{−1}(n, k) are displayed as triangular arrays for 1 ⩽ k ⩽ n ⩽ 8. Here, for immediate reference, we record these arrays in linearized form:

\begin{aligned}S_{−1}^{∗}(n, k) & = \begin{matrix}−1, & 1, & −1, & −1, & 1, & 1, & 1, & −1, & −2, & 1, & −1, & 1, \\ 3, & −2, & −1, & 1, & −1, & −4, & 3, & 3, & −1, & −1, & 1, & 5, \\ −4, & −6, & 3, & 1, & 1, & −1, & −6, & 5, & 10, & −6, & −4, & 1, & … \end{matrix} \\ S_{−1}(n, k) & = \begin{matrix}1, & 1, & −1, & 1, & −1, & −1, & 1, & −1, & −2, & 1, & 1, & −1, \\ −3, & 2, & 1, & 1, & −1, & −4, & 3, & 3, & −1, & 1, & −1, & −5, \\ 4, & 6, & −3, & −1, & 1, & −1, & −6, & 5, & 10, & −6, & −4, & 1, & … \end{matrix} \\ S̃_{−1}(n, k) & = \begin{matrix}1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 2, & 1, & 1, & 1, \\ 3, & 2, & 1, & 1, & 1, & 4, & 3, & 3, & 1, & 1, & 1, & 5, \\ 4, & 6, & 3, & 1, & 1, & 1, & 6, & 5, & 10, & 6, & 4, & 1, & … \end{matrix} \end{aligned}

2 Preliminaries

This section reviews some material to be used later in the paper.

2.1. Comtet Numbers. The following theorem, due to Comtet [2] greatly facilitates the analysis of many combinatorial arrays:

Theorem 2.1. Let D be an integral domain. If (u_{n})_{n⩾0} is a sequence in D and x is an indeterminate over D, then the following are equivalent characterizations of an array (U(n, k))_{n,k⩾0}:

U(n, k) = U(n − 1, k − 1) + u_{k}U(n − 1, k),  ∀ n, k ∈ ℙ,  (4)

with U(n, 0) = u_{0}^{n} and U(0, k) = δ_{0,k} ∀ n, k ∈ ℕ,

U(n, k) = ∑_{\begin{matrix}d_{0}+d_{1}+···+d_{k}=n−k \\ d_{i}∈ℕ \end{matrix}} u_{0}^{d_{0}}u_{1}^{d_{1}} ⋯ u_{k}^{d_{k}},  ∀ n, k ∈ ℕ,  (5)

∑_{n⩾0} U(n, k)x^{n} = \frac{x^{k}}{(1 − u_{0}x)(1 − u_{1}x) ⋯ (1 − u_{k}x)},  ∀ k ∈ ℕ,  (6)

and

x^{n} = ∑_{k=0}^{n} U(n, k)p_{k}(x),  ∀ n ∈ ℕ,  (7)

where p_{0}(x) := 1 and p_{k}(x) := (x − u_{0}) ⋯ (x − u_{k−1}) for k ∈ ℙ.

Proof. Straightforward algebraic exercise.

In what follows, we call the numbers U(n, k) the Comtet numbers associated with the sequence (u_{n})_{n⩾0}.

2.2 Partitions of a Set. A partition of a set S is a set of nonempty, pairwise disjoint subsets (called blocks) of S, with union S. For all n, k ∈ ℕ, let S(n, k) denote the number of partitions of [n] with k blocks. Then S(0, 0) = 1, S(n, 0) = S(0, k) = 0, ∀ n, k ∈ ℙ, and

S(n, k) = S(n − 1, k − 1) + kS(n − 1, k),  ∀ n, k ∈ ℙ,  (8)

S(n − 1, k − 1) enumerating those partitions in which n is the sole element of one of the blocks, and kS(n − 1, k) those in which the block containing n contains at least one other element of [n]. From (8) it follows that the numbers S(n, k), called Stirling numbers of the second kind, are the Comtet numbers associated with the sequence (0, 1, 2, …). Hence by Theorem 2.1

S(n, k) = ∑_{\begin{matrix}d_{1}+···+d_{k}=n−k \\ d_{i}∈ℕ \end{matrix}} 1^{d_{1}}2^{d_{2}} ⋯ k^{d_{k}},  ∀ n, k ∈ ℕ,

∑_{n⩾0} S(n, k)x^{n} = \frac{x^{k}}{(1 − x)(1 − 2x) ⋯ (1 − kx)}, ∀ k ∈ ℕ,

and

x^{n} = ∑_{k=0}^{n} S(n, k)x\underline{k}, ∀ n ∈ ℕ,

where x\underline{0} := 1 and x\underline{k} := x(x − 1) ⋯ (x − k + 1) for k ∈ ℙ.

The total number of partitions of [n] is given by the Bell number B_{n}, where

B_{n} = ∑_{k=0}^{n} S(n, k).

Clearly, B_{0} = 1, and

B_{n+1} = ∑_{k=0}^{n} \binom{n}{k}B_{k},

since \binom{n}{k}B_{k} enumerates those partitions of [n + 1] for which the size of the block containing the element n + 1 is n − k + 1.

2.3 Restricted Sums of Binomial Coefficients. As we have already noted in § 1, setting q = 1 and q = −1 in (1) yields the well known result

∑_{k even} \binom{n}{k} = ∑_{k odd} \binom{n}{k} = 2^{n−1}, ∀ n ∈ ℙ.

Here we recall a method for evaluating sums such as

∑_{k≡0 (mod 3)} \binom{n}{k}.

Let ω be either of the two complex cube roots of 1, e.g., ω = (−1 + i\sqrt{3})/2. Then

(1 + x)^{n} + (1 + ωx)^{n} + (1 + ω^{2}x)^{n} = ∑_{k=0}^{n} \binom{n}{k}x^{k} (1 + ω^{k} + ω^{2k})

= 3 ∑_{k≡0 (mod 3)} \binom{n}{k}x^{k}, (9)

since k ≡ 0 (mod 3) implies that 1 + ω^{k} + ω^{2k} = 3 and k ≡ 1 or 2 (mod 3) implies that 1 + ω^{k} + ω^{2k} = 1 + ω + ω^{2} = 0. Setting x = 1 in (9) yields

∑_{k≡0 (mod 3)} \binom{n}{k} = \frac{1}{3} (2^{n} + (1 + ω)^{n} + (1 + ω^{2})^{n}) .  (10)

3 Partition Statistics and q-Stirling Numbers

Let Π(n, k) denote the set of all partitions of [n] with k blocks. Given a partition π ∈ Π(n, k), let (E_{1}, …, E_{k}) be the unique ordered partition of [n] comprising the same blocks as π, arranged in increasing order of their smallest elements, and define statistics w^{∗}, w, and w̃ by

w^{∗}(π) := ∑_{i=1}^{k} i|E_{i}|,

w(π) := ∑_{i=1}^{k}(i − 1)|E_{i}| = w^{∗}(π) − n,

and

w̃(π) := ∑_{i=1}^{k}(i − 1)(|E_{i}| − 1) = w^{∗}(π) − n − \binom{k}{2}.

If elements of [n] are regarded as labels on n unit masses, then w^{∗}(π) is the moment about x = 0 of the mass configuration in which the masses with labels in E_{i} are placed at x = i. The statistics w(π) and w̃(π) admit of similar interpretations.

We wish to study the generating functions

S_{q}^{∗}(n, k) := ∑_{π∈Π(n,k)} q^{w^{∗}(π)},  (11)
S_{q}(n, k) := ∑_{π∈Π(n,k)} q^{w(π)} = q^{−n}S_{q}^{∗}(n, k),  (12)

and

S̃_{q}(n, k) := ∑_{π∈Π(n,k)} q^{w̃(π)} = q^{−n−\binom{k}{2}}S_{q}^{∗}(n, k).  (13)

Each of these polynomials furnishes a q-generalization of S(n, k), reducing to the latter when q = 1. As closely related as these q-Stirling numbers appear to be, it might be thought that one could carry out an analysis of any one of them, chosen arbitrarily, with properties of the others derived as easy corollaries. Interestingly, it turns out that each is best suited for elucidating a particular subset of their more or less common properties. We consider first the matter of recursive formulas.

Theorem 3.1. The q-Stirling numbers S_{q}^{∗}(n, k) are generated by the recurrence relation

S_{q}^{∗}(n, k) = q^{k}S_{q}^{∗}(n − 1, k − 1) + qk_{q}S_{q}^{∗}(n − 1, k),  ∀ n, k ∈ ℙ,  (14)

with S_{q}^{∗}(0, 0) = 1 and S_{q}^{∗}(n, 0) = S_{q}^{∗}(0, k) = 0, ∀ n, k ∈ ℙ.

Proof. The boundary conditions are obvious. To establish the recurrence (14), let

c(n, k, t) := |{π ∈ Π(n, k) : w^{∗}(π) = t}| .

Then,

c(n, k, t) = c(n − 1, k − 1, t − k) + ∑_{i=1}^{k} c(n − 1, k, t − i),  ∀ n, k ∈ ℙ.  (15)

For if w^{∗}(π) = t, with (E_{1}, …, E_{k}) being the ordered partition associated with π, then the number n ∈ [n] is either (i) in E_{k} alone (there are clearly c(n − 1, k − 1, t − k) such π’s) or (ii) in some E_{i}, where 1 ⩽ i ⩽ k, with at least one element of [n − 1] (there are clearly c(n − 1, k, t − i) such π’s). From (15) it follows that

S_{q}^{∗}(n, k) = ∑_{t} c(n, k, t)q^{t}

= ∑_{r} c(n − 1, k − 1, r)q^{r+k} + ∑_{i=1}^{k} q^{i} ∑_{r} c(n − 1, k, r)q^{r}

= q^{k}S_{q}^{∗}(n − 1, k − 1) + qk_{q}S_{q}^{∗}(n − 1, k).

Recurrence relations for S_{q}(n, k) and S̃_{q}(n, k) follow immediately from (14), along with (12) and (13), respectively. We have

S_{q}(n, k) = q^{k−1}S_{q}(n − 1, k − 1) + k_{q}S_{q}(n − 1, k),  ∀ n, k ∈ ℙ,  (16)

and

S̃_{q}(n, k) = S̃_{q}(n − 1, k − 1) + k_{q}S̃_{q}(n − 1, k),  ∀ n, k ∈ ℙ.  (17)

By (17), the numbers S̃_{q}(n, k) are the Comtet numbers associated with the sequence (n_{q})_{n⩾0}. By Theorem 2.1 it follows immediately that

S̃_{q}(n, k) = ∑_{\begin{matrix}d_{1}+···+d_{k}=n−k \\ d_{i}∈ℕ \end{matrix}}(1_{q})^{d_{1}}(2_{q})^{d_{2}} ⋯ (k_{q})^{d_{k}},  ∀ n, k ∈ ℕ,  (18)

∑_{n⩾0} S̃_{q}(n, k)x^{n} = \frac{x^{k}}{(1 − 1_{q}x)(1 − 2_{q}x) ⋯ (1 − k_{q}x)},  ∀ k ∈ ℕ,  (19)

and

x^{n} = ∑_{k=0}^{n} S̃_{q}(n, k)ϕ_{k}(x),  ∀ n ∈ ℕ,  (20)

where ϕ_{0}(x) := 1 and ϕ_{k}(x) := x(x − 1_{q}) ⋯ (x − (k − 1)_{q}), ∀ k ∈ ℙ.

Variants of (18)–(20) that hold for S_{q}(n, k) and S_{q}^{∗}(n, k) follow immediately from the relations S_{q}(n, k) = q^{\binom{k}{2}}S̃_{q}(n, k) and S_{q}^{∗}(n, k) = q^{n}S_{q}(n, k). To cite a few examples, we have

∑_{n⩾0} S^{∗}(n, k)x^{n} = \frac{q^{\binom{k+1}{2}}x^{k}}{(1 − qx)(1 − qx − q^{2}x) ⋯ (1 − qx − ⋯ − q^{k}x)},

∀ k ∈ ℕ,

and

x^{n} = ∑_{k=0}^{n} S_{q}(n, k)ψ_{k}(x) = ∑_{k=0}^{n} S_{q}^{∗}(n, k)ψ_{k} (\frac{x}{q}) ,  (21)

where ψ_{k}(x) := q^{−\binom{k}{2}}ϕ_{k}(x).

Using the method of linear functionals [5, pp. 89–90] one can derive from (21) the recurrence [8, Theorem 5.4]

S_{q}(n + 1, k) = ∑_{j=0}^{n} \binom{n}{j}q^{j}S_{q}(j, k − 1),  ∀ n ∈ ℕ, k ∈ ℙ,  (22)

from which the variant recurrences

S_{q}^{∗}(n + 1, k) = q^{n+1} ∑_{j=0}^{n} \binom{n}{j}S_{q}^{∗}(j, k − 1),  ∀ n ∈ ℕ, k ∈ ℙ,

and

S̃_{q}(n + 1, k) = ∑_{j=0}^{n} \binom{n}{j}q^{j−k+1}S̃_{q}(j, k − 1),  ∀ n ∈ ℕ, k ∈ ℙ  (23)

follow immediately.

Summing the q-Stirling numbers S_{q}^{∗}(n, k), S_{q}(n, k) and S̃_{q}(n, k) over k yields the respective q-Bell numbers B_{q}^{∗}(n), B_{q}(n), and B̃_{q}(n). From (22) it follows that

B_{q}(n + 1) = ∑_{j=0}^{n} \binom{n}{j}q^{j}B_{q}(j),  ∀ n ∈ ℕ.  (24)

Since B_{q}^{∗}(n) = q^{n}B_{q}(n), the recurrence (24) yields

B_{q}^{∗}(n + 1) = q^{n+1} ∑_{j=0}^{n} \binom{n}{j}B_{q}^{∗}(j),  ∀ n ∈ ℕ.  (25)

Due to the factor q^{−k} in (23), we do not get any recurrence for B̃_{q}(n) analogous to (24) and (25), this being the single exception to the general parallelism between properties of the three q-Stirling numbers under consideration. The uniqueness of B̃_{q}(n) is further manifested when q = −1, as we shall see in the next section.

4 The Case q = −1

In this section we derive simple expressions for the foregoing q-Stirling and q-Bell numbers when q = −1.

Theorem 4.1. The number S̃_{−1}(n, k) is given by the formula

S̃_{−1}(n, k) = \binom{n − ⌊\frac{k}{2}⌋ − 1}{⌊\frac{k−1}{2}⌋},  1 ⩽ k ⩽ n.  (26)

Proof. Note that

i_{q}|_{q=−1} = ω_{i} := \begin{cases}1, & if i is odd; \\ 0, & if i is even. \end{cases}

Hence by (18), if 1 ⩽ m ⩽ ⌊n/2⌋,

S̃_{−1}(n, 2m) = ∑_{\begin{matrix}d_{1}+d_{3}+···+d_{2m−1}=n−2m \\ d_{i}∈ℕ \end{matrix}} 1 = \binom{n − m − 1}{m − 1},  (27)

since the number of sequences (t_{1}, …, t_{m}) of nonnegative integers summing to s is \binom{s+m−1}{m−1} [7, p. 15]. Similarly, if 0 ⩽ m ⩽ ⌊(n − 1)/2⌋,

S̃_{−1}(n, 2m + 1) = \binom{n − m − 1}{m}.  (28)

Formula (26) incorporates (27) and (28).

In tabulating the numbers S̃_{−1}(n, k) it is of course more efficient to use the recurrence

S̃_{−1}(n, k) = S̃_{−1}(n − 1, k − 1) + ω_{k}S̃_{−1}(n − 1, k),

representing the case q = −1 of (17).

Let F_{0} = F_{1} = 1, with F_{n} = F_{n−1} + F_{n−2} if n ⩾ 2. As is well known,

F_{n} = ∑_{m=0}^{⌊n/2⌋} \binom{n − m}{m},  ∀n ∈ ℕ.  (29)

Theorem 4.2. For all n ∈ ℕ,

B̃_{−1}(n) := ∑_{k=0}^{n} S̃_{−1}(n, k) = F_{n}.  (30)

Proof. It is easy to check that (30) holds for n = 0, 1. If n ⩾ 2, then by (27), (28), and (29),

B̃_{−1}(n) = ∑_{m=0}^{⌊(n−1)/2⌋} \binom{n − m − 1}{m} + ∑_{m=1}^{⌊n/2⌋} \binom{n − m − 1}{m − 1}

= ∑_{m=0}^{⌊(n−1)/2⌋} \binom{(n − 1) − m}{m} + ∑_{m=0}^{⌊(n−2)/2⌋} \binom{(n − 2) − m}{m}

= F_{n−1} + F_{n−2} = F_{n}.

From (26) and the fact that S_{q}^{∗}(n, k) = q^{\binom{k}{2}+n}S̃_{q}(n, k), we have

S_{−1}^{∗}(n, k) = (−1)^{\binom{k}{2}+n}\binom{n − ⌊\frac{k}{2}⌋ − 1}{⌊\frac{k−1}{2}⌋}, 1 ⩽ k ⩽ n.

On the other hand, the Bell numbers B_{−1}^{∗}(n) are quite different from the numbers B̃_{−1}(n). Theorem 4.3. For all n ∈ ℕ,

B_{−1}^{∗}(n) := ∑_{k=0}^{n} S_{−1}^{∗}(n, k) = \begin{cases}1, & if n ≡ 0 (mod 3); \\ −1, & if n ≡ 1 (mod 3); \\ 0, & if n ≡ 2 (mod 3). \end{cases}  (31)

Proof. Noting that B_{−1}^{∗}(0) = 1, we prove (31) by induction on n. In what follows

b_{r}(n) := ∑_{j≡r (mod 3)} \binom{n}{j}.

From (25) with q = −1, we have

B_{−1}^{∗}(n + 1) = (−1)^{n+1} ∑_{j=0}^{n} \binom{n}{j}B_{−1}^{∗}(j) = (−1)^{n+1}b_{0}(n) + (−1)^{n}b_{1}(n)

= (−1)^{n+1}b_{0}(n) + (−1)^{n}b_{0}(n − 1) + (−1)^{n}b_{1}(n − 1).

Similarly, B_{−1}^{∗}(n) = (−1)^{n}b_{0}(n − 1) + (−1)^{n−1}b_{1}(n − 1), and so

B_{−1}^{∗}(n + 1) = (−1)^{n+1}b_{0}(n) + 2(−1)^{n}b_{0}(n − 1) − B_{−1}^{∗}(n)

= \frac{1}{3} [ω^{2n−1} − ω^{2n−2} + ω^{n+1} − ω^{n−1}] − B_{−1}^{∗}(n), (32)

by (10), where ω is either of the two complex cube roots of 1. Taking n + 1 = 3m, 3m + 1, and 3m + 2, respectively, in (32) yields

B_{−1}^{∗}(3m) = 1 − B_{−1}^{∗}(3m − 1) = 1,
B_{−1}^{∗}(3m + 1) = 0 − B_{−1}^{∗}(3m) = −1,  and
B_{−1}^{∗}(3m + 2) = −1 − B_{−1}^{∗}(3m + 1) = 0.

It is easy to check that one can write (31) more compactly as

B_{−1}^{∗}(n) = \frac{1}{1 − ω} ω^{n} − \frac{ω}{1 − ω} ω^{2n},

from which we get the nice exponential generating function

∑_{n=0}^{∞} B_{−1}^{∗}(n)\frac{x^{n}}{n!} = \frac{1}{1 − ω} e^{ωx} − \frac{ω}{1 − ω} e^{ω^{2}x}.  (33)

From (26) and the fact that S_{q}(n, k) = q^{\binom{k}{2}}S̃_{q}(n, k), we have

S_{−1}(n, k) = (−1)^{\binom{k}{2}}\binom{n − ⌊\frac{k}{2}⌋ − 1}{⌊\frac{k−1}{2}⌋}, 1 ⩽ k ⩽ n.

By (12),

B_{−1}(n) := ∑_{k=0}^{n} S_{−1}(n, k) = (−1)^{n}B_{−1}^{∗}(n),

and so by (31)

B_{−1}(n) = \begin{cases}(−1)^{n}, & if n ≡ 0 (mod 3); \\ (−1)^{n+1}, & if n ≡ 1 (mod 3); \\ 0, & if n ≡ 2 (mod 3), \end{cases}  (34)

and by (33)

∑_{n=0}^{∞} B_{−1}(n)\frac{x^{n}}{n!} = \frac{1}{1 − ω} e^{−ωx} − \frac{ω}{1 − ω} e^{−ω^{2}x}.  (35)

In a paper which showed how the numbers S_{−1}(n, k) and B_{−1}(n) arise in the study of fermionic oscillators, Schork [6] posed the problem of finding a closed form and a generating function for the numbers B_{−1}(n). Formulas (34) and (35) furnish solutions to Schork’s problem.

To conclude this section we remark that our list of partition statistics might have been rounded out to include the statistic

ŵ(π) := ∑_{i=1}^{k} i(|E_{i}| − 1) = w̃(π) + n − k,

with generating function

Ŝ_{q}(n, k) := ∑_{π∈Π(n,k)} q^{ŵ(π)} = q^{(n−k)}S̃_{q}(n, k).  (36)

Formula (36) and Theorem 4.1 yield an easy evaluation of Ŝ_{−1}(n, k). As for

B̂_{−1}(n) := ∑_{k=0}^{n} Ŝ_{−1}(n, k),

we have B̂_{−1}(0) = B̂_{−1}(1) = 1, B̂_{−1}(2) = 0, and

B̂_{−1}(n) = (−1)^{n−1}F_{n−3},  ∀n ⩾ 3,  (37)

the proof of which we leave to interested readers.

5 Bijective Proofs

We conclude by returning to the opening theme of this paper. If G(I, Δ; −1) = 0, then, as already noted, |Δ_{0}| = |Δ_{1}|, where Δ_{i} = {δ ∈ Δ : I(δ) ≡ i (mod 2)}. One gains a deeper understanding (and bijective proof) of such results by identifying an I-parity changing involution of Δ. For the statistic |S| in (1), the map

S ↦ \begin{cases}S ∪ \{1\}, & if 1 ∉ S; \\ S − \{1\}, & if 1 ∈ S, \end{cases}

furnishes such an involution. For the statistic i(σ) in (2), switching positions of the elements 1 and 2 (or of k and k + 1, for any fixed k) in a permutation furnishes such an involution.

A similar task arises when (3) is nonzero. Suppose, for example, that G(I, Δ; −1) = c > 0. Here one wishes to identify a subset Δ^{+} of Δ_{0}, with |Δ^{+}| = c, and an I-parity changing involution of Δ − Δ^{+}.

My student, Mark Shattuck, has recently succeeded in finding such bijective proofs of formulas (26), (30), (31), (34), and (37). Details will appear in a forthcoming paper.

6 Tables

Table 1: The numbers S_{−1}^{∗}(n, k) for 1 ⩽ k ⩽ n ⩽ 8.

k = 1 2 3 4 5 6 7 8
n = 1  −1
2  1 −1
3  −1 1 1
4  1 −1 −2 1
5  −1 1 3 −2 −1
6  1 −1 −4 3 3 −1
7  −1 1 5 −4 −6 3 1
8  1 −1 −6 5 10 −6 −4 1

Table 2: The numbers S_{−1}(n, k) for 1 ⩽ k ⩽ n ⩽ 8.

k = 1 2 3 4 5 6 7 8
n = 1  1
2  1 −1
3  1 −1 −1
4  1 −1 −2 1
5  1 −1 −3 2 1
6  1 −1 −4 3 3 −1
7  1 −1 −5 4 6 −3 −1
8  1 −1 −6 5 10 −6 −4 1

Table 3: The numbers S̃_{−1}(n, k) for 1 ⩽ k ⩽ n ⩽ 8.

k = 1 2 3 4 5 6 7 8
n = 1  1
2  1 1
3  1 1 1
4  1 1 2 1
5  1 1 3 2 1
6  1 1 4 3 3 1
7  1 1 5 4 6 3 1
8  1 1 6 5 10 6 4 1

References

[1] L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University (1968), 8--15.

[2] L. Comtet, Nombres de Stirling généraux et fonctions symmetriques, C. R. Acad. Sci. Paris Série A 275 (1972), 747--750.

[3] P. Edelman, R. Simion, and D. White, Partition statistics on permutations, Discrete Math. 99 (1992), 62--68.

[4] S. Milne, A q-analogue of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. Amer. Math. Soc. 245 (1978), 89--118.

[5] G.-C. Rota, The number of partitions of a set, Amer. Math. Monthly 71 (1964), 498-- 504.

[6] M. Schork, On the combinatorics of normal ordering bosonic operators and deformations of it, J. Phys. A: Math. Gen. 36 (2003), 4651--4665.

[7] R. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986.

[8] C. Wagner, Generalized Stirling and Lah numbers, Discrete Math. 160 (1996), 199--218.

[9] M. Wachs and D. White, p, q-Stirling numbers and partition statistics, J. Combin. Theory A 56 (1991), 27--46.

2000 Mathematics Subject Classification: Primary 11B73; Secondary 11B39. Keywords: partition statistics, q-Stirling numbers, q-Bell numbers, Fibonacci numbers.

(Concerned with sequence \underline{A000045}.)

Received August 1, 2003; revised version received January 8, 2004. Published in Journal of Integer Sequences, February 9 2004.

Return to Journal of Integer Sequences home page.