[Paper Review] Card deals, lattice paths, abelian words and combinatorial identities

David Callan|ArXiv.org|Dec 28, 2008

semigroups and automata theory3 references3 citations

TL;DR

This paper provides combinatorial interpretations for three families of identities involving Franel and Apéry numbers, using derangement-type card deals, labeled lattice paths, and abelian matrices over a 3-letter alphabet. It establishes a bijection between Barrucand card deals and abelian matrices, and uses these constructs to give direct, bijective proofs of identities such as $\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}^2 = \sum_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}\sum_{j=0}^{k}\binom{k}{j}^3$, linking them to lattice paths and matrix statistics.

ABSTRACT

We give combinatorial interpretations of several related identities associated with the names Barrucand, Strehl and Franel, including one for the Apery numbers. The combinatorial constructs employed are derangement-type card deals as introduced in a previous paper on Barrucand's identity, labeled lattice paths and, following a comment of Jeffrey Shallit, abelian words over a 3-letter alphabet.

Motivation & Objective

• To provide combinatorial interpretations for identities related to Franel and Apéry numbers using constructive combinatorial models.
• To establish a bijection between Barrucand n-deal card configurations and 2×n abelian matrices over {1,2,3}, linking them to abelian squares.
• To unify multiple equivalent combinatorial expressions in identity (3) through three equinumerous constructs: card deals, lattice paths, and matrices.
• To offer direct, bijective proofs of identities involving sums of cubes of binomial coefficients and weighted path counts.
• To resolve a question posed by Jeffrey Shallit on abelian word interpretations for Barrucand-type identities.

Proposed method

• Introduces 'Hanna n-deals' as a generalization of Barrucand's derangement-type card deals, where cards of three colors are dealt such that no player receives their own color.
• Defines labeled lattice paths with steps U (up), D (down), and F (flat) with labels in {1,2,3}, and introduces statistics X (non-1-labeled matching UD pairs) and Y (flat steps with label >2).
• Constructs a 1-to-1 correspondence between Barrucand n-deals and 2×n abelian matrices (representing abelian squares), using column types to encode card distribution and hand ownership.
• Uses the transformation rules: U→UU, D→DD, F₁→UD, F₂→DU to map labeled paths to Dyck-like paths and reverse the process, preserving path length and step counts.
• Applies the Chu–Vandermonde identity to simplify sums over statistics X and Y, proving the equivalence of expressions in identity (3).
• Interprets each summand in identity (3) combinatorially: $\binom{n}{k}\binom{2k}{k}2^k$ counts Hanna n-deals by red hand size, $\binom{n}{k}\binom{2n-k}{n}3^k$ by X+Y=k, $\binom{n}{k}^2 4^k$ by 1s in each matrix row, and $\binom{n}{2k}\binom{2k}{k}4^k 5^{n-2k}$ by number of upsteps in paths.

Experimental results

Research questions

• RQ1 Can Barrucand’s identity be given a direct combinatorial interpretation via abelian matrices over a 3-letter alphabet?
• RQ2 How can the Franel identity $\sum_{k=0}^{n}\binom{n}{k}^3 = \sum_{k=0}^{n}\binom{n}{k}^2\binom{2k}{n}$ be interpreted combinatorially using card deals or lattice paths?
• RQ3 What is the combinatorial meaning of the multi-expression identity (3), which equates four different sums involving binomial coefficients and powers?
• RQ4 Is there a natural bijection between Barrucand n-deals, labeled lattice paths with specific statistics, and 2×n abelian matrices?
• RQ5 Can the sum $\sum_{j=0}^{k}\binom{k}{j}^3$ in Barrucand’s identity be interpreted via abelian matrix column types?

Key findings

• A bijection is established between Barrucand n-deals and 2×n abelian matrices over {1,2,3}, with each matrix column type corresponding to a specific card distribution pattern.
• The identity $\sum_{k=0}^{n}\binom{n}{k}\sum_{j=0}^{k}\binom{k}{j}^3 = \sum_{k=0}^{n}\binom{n}{k}^2\binom{2k}{k}$ is interpreted as counting the same set of abelian matrices via two different statistics: total cards in red’s hand and number of distinct denominations.
• The sum $\binom{n}{k}\binom{2k}{k}2^k$ counts Hanna n-deals with k cards in red’s hand, where the $2^k$ factor arises from choosing red cards in blue’s hand.
• The expression $\binom{n}{k}\binom{2n-k}{n}3^k$ counts Hanna n-paths with X+Y=k, where X is the number of non-1-labeled matching UD pairs and Y is the number of flat steps with label >2.
• The sum $\binom{n}{k}^2 4^k$ counts 2×n abelian matrices with n−k ones in each row, corresponding to Hanna n-deals with n−k denominations in red’s hand.
• The sum $\binom{n}{2k}\binom{2k}{k}4^k 5^{n-2k}$ counts Hanna n-paths with exactly k upsteps, where flat steps are labeled in 5 ways and slanted steps are labeled in 4 ways.