[Paper Review] Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients

Maciej Dziemiańczuk|ArXiv.org|Jan 11, 2009

Advanced Combinatorial Mathematics12 references5 citations

TL;DR

This paper establishes a novel combinatorial interpretation of $N(\alpha)$-nomial coefficients—generalizations of binomial and Gaussian coefficients—as counting labeled bipartite $\alpha$-multigraphs and $k$-colored $\alpha$-multigraphs. It further proves that the unsigned first row of the inversion matrix of these coefficients equals the number of labeled directed acyclic $\alpha$-multigraphs, providing a closed-form enumeration via signed sums of $N(\alpha)$-nomial coefficients.

ABSTRACT

F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation between the number of k-colored a-multigraphs and multi N(a)-nomial coefficients is established. We also prove that the unsigned values of the first row of inversion matrix for N(a) -nomial coefficients considered here are equal to the numbers of directed acyclic a-multigraphs with n nodes.

Motivation & Objective

• To extend the combinatorial interpretation of $F$-nomial coefficients beyond general cobweb-admissible sequences to specific $N(\alpha)$-tiling sequences.
• To establish a direct correspondence between $N(\alpha)$-nomial coefficients and the number of labeled bipartite $\alpha$-multigraphs with $k$ vertices in one part.
• To derive a formula for the number of $k$-colored $\alpha$-multigraphs using $N(\alpha)$-nomial coefficients.
• To prove that the unsigned entries of the first row of the inversion matrix of $N(\alpha)$-nomial coefficients count labeled directed acyclic $\alpha$-multigraphs with $n$ vertices.
• To unify and generalize existing results on multigraph enumeration using a parameterized family of $F$-nomial coefficients derived from $N(\alpha)$ sequences.

Proposed method

• Define $N(\alpha)$ as a cobweb-tiling sequence with $n_{F} = n \cdot \alpha^{n-1}$, forming the basis for $N(\alpha)$-nomial coefficients.
• Use the standard $F$-nomial coefficient formula $\binom{n}{k}_F = \frac{n_F^{– k}}{k_F!}$, where $n_F^{– k}$ denotes the falling factorial over the sequence $F$.
• Establish that the number of labeled bipartite $\alpha$-multigraphs $G(\alpha,n,k)$ is $\binom{n}{k}_F \cdot \alpha^{k(n-k)}$, linking it to $N(\alpha)$-nomial coefficients.
• Apply the inclusion-exclusion principle to count directed acyclic $\alpha$-multigraphs by considering vertices with in-degree zero (out-points), leading to a recursive formula.
• Derive the inversion formula for $N(\alpha)$-nomial coefficients, showing $\binom{n}{0}_F^{-1} = \sum_{s=1}^n (-1)^s \sum_{k_1+\cdots+k_s=n \atop k_i \geq 1} \binom{n}{k_1,\dots,k_s}_F$.
• Prove that the number of labeled acyclic $\alpha$-multigraphs $A_\alpha(n)$ equals $|\binom{n}{0}_F^{-1}| = (-1)^n \binom{n}{0}_F^{-1}$, using recursive expansion and reindexing of sums over compositions.

Experimental results

Research questions

• RQ1 How can $N(\alpha)$-nomial coefficients be combinatorially interpreted in terms of multigraph structures?
• RQ2 What is the explicit formula for the number of $k$-colored $\alpha$-multigraphs, and how does it relate to $N(\alpha)$-nomial coefficients?
• RQ3 Can the number of labeled directed acyclic $\alpha$-multigraphs be expressed via the inversion matrix of $N(\alpha)$-nomial coefficients?
• RQ4 What is the role of the inclusion-exclusion principle in enumerating acyclic multigraphs with $\alpha$-multiple edges?
• RQ5 How does the structure of $N(\alpha)$ sequences enable a generalization of binomial and Gaussian coefficient interpretations to multigraphs?

Key findings

• The number of labeled bipartite $\alpha$-multigraphs with $n$ vertices and $k$ in one part is $\beta_{\alpha,n,k} = \binom{n}{k}_{N(\alpha)} \cdot \alpha^{k(n-k)}$, directly linking it to $N(\alpha)$-nomial coefficients.
• The number of $k$-colored $\alpha$-multigraphs is explicitly given by $\binom{n}{k}_{N(\alpha)} \cdot \alpha^{k(n-k)}$, extending the interpretation to colored variants.
• The unsigned first row of the inversion matrix of $N(\alpha)$-nomial coefficients equals the number of labeled directed acyclic $\alpha$-multigraphs with $n$ vertices.
• The number of such acyclic multigraphs is $A_\alpha(n) = (-1)^n \binom{n}{0}_{N(\alpha)}^{-1}$, with $A_\alpha(0) = 1$, providing a closed-form expression.
• The derivation uses recursive inclusion-exclusion over out-points in acyclic multigraphs, leading to a sum over compositions of $n$ with signed $N(\alpha)$-nomial coefficients.
• The result generalizes known cases such as $\alpha=2$, where the formula reduces to previously studied enumeration of bipartite and acyclic graphs using $F$-nomial coefficients.