Combinatorial Identities by Way of Wilf's Multigraph Model

For many families of combinatorial objects, a construction of Wilf (1977) allows the members of the family to be viewed as paths in a directed multigraph. Introducing a partition of these paths generates a number of known, but hitherto disparate, combinatorial identities. We include several examples.


Introduction
We begin with a well-known pair of identities, one satisfied by each type of Stirling number.For 1 ≤ t ≤ n, Stirling numbers of the first kind are defined combinatorially by For 1 ≤ t < n, these numbers satisfy the identity For example, For 1 ≤ t ≤ n, Stirling numbers of the second kind are defined by n t := The number of t-block partitions of [n]. (1.4) For 1 ≤ t < n, these numbers satisfy the identity (1.5) An example of this second identity is given by Combinatorial proofs of these identities can be found in [6], but we submit that the proper setting for understanding these identities as well as several others is one discovered by Wilf.In [7], Wilf provides a way to associate a directed multigraph to any family of combinatorial objects, provided only that the family admits a method of constructing the larger objects from the smaller ones.The set of all t-cycle permutations of [n] (for all t and n) constitutes such a family, as does the set of all t-block partitions of [n]; these are among the examples that Wilf gives in his paper.In this construction, the multigraph models the entire family, while individual objects correspond to certain paths in the graph; it turns out that partitioning the paths by vertex sequence generates a number of identities, including (1.2) and (1.5).
The formula that results from the partitioning unifies all of the identities we consider in this paper.Section 2 consists of a brief account of Wilf 's construction and the most general version of our formula.In Sections 3 and 4, we specialize to contexts in which the multigraphs come with additional structure.In each of these two sections, the general formula assumes a more computational form, which then leads directly to a number of combinatorial identities.We discuss several examples, including (1.2) and (1.5).Finally, in Section 5, we discuss one identity that falls within the scope of the general formula but outside that of Sections 3 and 4.

Combinatorial family graphs
Definition 2.1.A combinatorial family graph G is a directed multigraph on a (usually infinite) vertex set V = V (G) satisfying the following conditions.
(a) Every vertex has finite indegree and finite outdegree.(b) There is a unique vertex τ ∈ V (the "terminal vertex") of outdegree zero; for every other vertex v, the outdegree of v, ρ o (v), is positive.(c) G has no paths of infinite length (so, in particular, G is acyclic).(d) For each v ∈ V , except for τ, the edges with initial vertex v are consecutively numbered (1,...,ρ o (v)).
Wilf 's numbering of the edges differs slightly from ours, but in all other respects, this definition is that found in [7].
From (a) and (c), it follows that only finitely many paths originate from any fixed vertex.For any vertex v, let b(v) denote the number of paths from v to τ (b(τ) = 1); these paths are the combinatorial objects of order v.
Since G is a multigraph, there will generally be multiple paths that follow the same vertex sequence.Partitioning the set of all paths v → τ by vertex sequence then gives our first theorem.

Application to restricted binomial families
Most of the instances of combinatorial family graphs that we treat here belong to a subclass of combinatorial family graphs; in this subclass, we get useful special forms of (2.1).
The following definition of a binomial family is essentially the one in [7], although there are a few minor differences.

Definition 3.1.
A binomial family is a combinatorial family graph for which the following hold.
(a) For some t 0 ≥ 0, V is a set of lattice points in the plane, namely, In this section, we consider the simplest case, in which ψ(n,t) = 1 for all n ≥ t ≥ (t 0 + 1).We call such binomial families restricted.
We now establish a computational form of (2.1) for restricted binomial families.For a vertex (n,t) in a restricted binomial family G, fix one vertex sequence from (n,t) to τ = (t 0 ,t 0 ).Such a sequence contains a total of (n − t 0 ) moves, with (t − t 0 ) of them being diagonal and (n − t) of them being horizontal.The sequence can thus be indexed by the set of x-coordinates of vertices that occur to the right of the horizontal edges.For example, the vertex sequence will be indexed by the set S = {2, 4,5,7}.The set S can now be used to find the number of paths with this vertex sequence.Since ψ(x, y) = 1 for all x ≥ y ≥ (t 0 + 1), the number of paths which follow this vertex sequence is given by where (k i ,t i ) is the vertex in the sequence with x-coordinate k i .(Note that there is one path when n = t.Here and below, "1" is the correct value for any empty product.)Furthermore, if we assume that the elements of S have been arranged in increasing order ( Summing this expression over all possible vertex sequences from (n,t) to (t 0 ,t 0 ) then gives the following theorem.
Theorem 3.2.Let G be a restricted binomial family, and let (n,t Identities (1.2) and (1.5) are the instances of Theorem 3.2 that hold in the restricted binomial family of permutations (resp., partitions).These families appear in [7], but for completeness, we include them here.
ensures that b(n,t) = [ n t ] for all (n,t) ∈ V .Moreover, the standard constructive proof of (3.6) gives the following bijection between the paths (n,t) → τ and the t-cycle permutations of [n].Given a path ω : (n,t) → τ, make the associated permutation σ as follows.
Observe that (1.2) is the instance of (3.5) in this family.Since φ(n,t) = (n − 1), we have ensures that b(n,t) = { n t } for all (n,t) ∈ V .Moreover, the standard constructive proof of (3.8) gives a bijection between the paths (n,t) → τ and the t-block partitions of [n].For completeness, again, we repeat the proof here, phrasing it so as to make the bijection clear.Given a path ω : (n,t) → τ, make the associated partition π as follows.
(I) If the first edge of ω is horizontal edge from (n,t) to (n − 1,t), where 1 ≤ ≤ φ(n,t) = t, then choose (recursively) a t-block partition of [n − 1], and insert n in the block with the th smallest minimal element (so that {n} will not be a singleton of π).(II) If the first edge of ω is the edge (n,t) → (n − 1,t − 1), choose (recursively) a (t − 1)block permutation of [n − 1], and adjoin {n} as a singleton.Observe that (1.5) is the instance of (3.5) in this family.Since φ(n,t) = t, we have (3.9) In this example, summand Next, we apply Theorem 3.2 in a restricted binomial family introduced by Calabi and Wilf [1].
Example 3.5.For 0 ≤ t ≤ n and for q, a power of a prime, define n t q := The number of t-dimensional subspaces of F (n) q , (3.10) where F q is the finite field with q elements.These numbers satisfy the well-known recurrence (3.11) In [1], Calabi and Wilf provide a constructive proof of (3.11), which we summarize here.Their proof capitalizes on the fact that for 1 ≤ t ≤ n, each t-dimensional subspace of F (n)   q is the row space of a unique t × n matrix over F q in reduced row echelon form (rref).In the top line of (3.11), the first term is the number of t × n rref matrices for which (t,n) is a pivot position, and the second term is the number of such matrices for which it is not.This recurrent construction gives an associated restricted binomial family G where the vertex set is given by V = {(n, t) : Applying (3.5) here gives, for n ≥ t + 1, where summand q u1+u2+•••+un−t is the number of t × n rref matrices with pivots in columns The Calabi/Wilf construction and formula (3.12) appear in [6].

Application to unrestricted binomial families
We now consider general binomial families, without the restriction ψ ≤ 1.The computational form of (2.1) is obtained as much as (3.5) was, although the formula is messier.Again, consider a fixed vertex sequence from (n,t) to τ = (t 0 ,t 0 ).Proceeding as before, we index this sequence by the same set S, but we now need to use S to compute the ycoordinate from the x-coordinate for every vertex (x, y) in the sequence, not just for those vertices with x-coordinates in S. The function which does this is the function Table 4.1 The function f S is computed in Table 4.1.Note that f S does correctly find the ycoordinate of each vertex from its x-coordinate.This function now gives us the analogue of (3.8) for general binomial families.The number of paths that follow the vertex sequence indexed by S is given by where S := [t 0 + 1,n] \ S. Summing over all possible subsets S gives the following theorem.Note that b(n,n) = n x=t0 ψ(x,x).Example 4.2.Formula (4.5) can be applied to a binomial family, introduced in [3], that models the following sets of trees.Definition 4.3.For 1 ≤ t ≤ (n − 2), let T(n,t) be the set of all trees on n labeled vertices, where vertices {1, ...,t} all have degree ≥ 2 and vertices {t + 1,...,n} are all leaves.(It is convenient to ignore the tree on one vertex and the tree on two vertices.)The triangle of numbers {|T(n, t)|} is the sequence A019538 in [5] and satisfies the recurrence relation below: Equation (4.6) can be established by a recurrent construction of the sets T(n,t).
(I) Given a tree -∈ T(n − 1,t) [for t ≤ (n − 3)] and an integer j, 1 ≤ j ≤ t, construct tree in T(n,t) by making vertex n adjacent to vertex j.If this is done for each choice ofand j, each tree in T(n,t) for which the vertex adjacent to n has degree ≥ 3 will be constructed exactly once.(II) Given a tree -∈ T(n − 1,t − 1) and an integer j, 1 ≤ j ≤ t, construct tree in T(n,t) by exchanging the labels of vertices j and t inmaking vertex n adjacent to the vertex now labeled j.If this is done for each choice ofand j, each tree in T(n,t) for which the vertex adjacent to n has degree = 2 will be constructed exactly once.We can now construct the associated binomial family; in order to do so, we must shift the graph two units to the left.Put V = {(n, t) : n ≥ t ≥ 1}, so that τ = (1,1).For n ≥ (t + 1) ≥ 2, put φ(n,t) = t; for n ≥ t ≥ 2, put ψ(n,t) = t.The trees in T(n,t) are then in bijection with the paths (n − 2,t) → (1,1), so that T(n,t) = b(n − 2,t).The well-known Prüfer bijection between labeled trees and sequences (e.g., described in [4]) induces a bijection between T(n,t) and the set of surjections from [n − 2] onto [t].
There is a also a straightforward bijection ←→ t-block partitions of [n − 2] × S t .(4.9) Composing these two bijections allows a variant of Example 3.4 to furnish another construction for (4.8), but the two recurrent constructions are not the same.