[Paper Review] Proofs of some binomial identities using the method of last squares

Mark Shattuck, Tamás Waldhauser|arXiv (Cornell University)|Jul 6, 2011

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper provides combinatorial proofs for a family of binomial identities involving sums of products of binomial coefficients that lack closed forms. Using tiling interpretations of dominos and colored squares on a 1×m board, the authors establish a bijection between different combinatorial structures, proving that five distinct expressions—S_{m,r}, T_{n,r}, U_{n,r}, V_{n,r}, and W_{n,r}—are equal for n = m−1−r. The key contribution is a closed-form-like expression W_{n,r} with a constant number of terms, revealing a hidden structure in otherwise intractable sums.

ABSTRACT

We give combinatorial proofs for some identities involving binomial sums that have no closed form.

Motivation & Objective

• To provide combinatorial proofs for binomial identities that lack closed-form expressions, particularly sums involving products of binomial coefficients.
• To establish the equality of five distinct combinatorial expressions: S_{m,r}, T_{n,r}, U_{n,r}, V_{n,r}, and W_{n,r} for n = m−1−r.
• To demonstrate that W_{n,r} serves as a simplified representation with a fixed number of terms, despite the original sums having variable-length summations.
• To develop a novel bijection between tiling configurations to prove the identity T_{n,r} = W_{n,r}, resolving a nontrivial parity-based discrepancy via an 'almost bijection' with a single fixed point.

Proposed method

• Modeling the sum S_{m,r} as the number of tilings of a 1×m board with r dominos and m−2r squares, where the first cell is covered by a black square and the last squares' colors are tracked.
• Defining three additional tiling models: T_{n,r} counts tilings with three types of squares (white, black, decorated), U_{n,r} uses a generating function approach with weights, and V_{n,r} uses a recursive decomposition based on positions of black squares.
• Establishing a bijection between the tiling models of S_{m,r} and T_{n,r} via a shift in indexing (n = m−1−r), showing structural equivalence through combinatorial mapping.
• Introducing a conjugation map on special tiling subsets (B_{n,r}^{+,o} and B_{n,r}^{-,e}) to prove Lemma 2.6, which establishes the parity-dependent difference of ±1.
• Using the conjugation map to prove Proposition 2.7, showing that T_{n,r} = W_{n,r} by expressing T_{n,r} as the size of even-weight tilings plus a correction term (−1)^{r+1}.
• Deriving the closed-form-like expression W_{n,r} = 2^{n−r}∑_{k=0}^{⌊r/2⌋} binom{n−2−2k}{r−2k} + (−1)^{r+1}, where the sum counts tilings with black squares placed in specific intervals.

Experimental results

Research questions

• RQ1 Can combinatorial proofs be constructed for binomial sums that lack hypergeometric closed forms?
• RQ2 What is the structural reason behind the equality of five seemingly different combinatorial expressions involving binomial coefficients?
• RQ3 Why does the expression W_{n,r} stand out as having a constant number of terms regardless of n, unlike the other sums?
• RQ4 How can a parity-dependent correction term (−1)^{r+1} arise naturally in a tiling-based counting problem?
• RQ5 Is there a bijection that explains the identity T_{n,r} = W_{n,r} despite the apparent asymmetry in their definitions?

Key findings

• The five expressions S_{m,r}, T_{m−1−r,r}, U_{m−1−r,r}, V_{m−1−r,r}, and W_{m−1−r,r} are all equal for 0 ≤ r ≤ m/2 − 1, establishing a deep combinatorial identity.
• The sum T_{n,r} = ∑_{j=r+1}^n binom{n}{j} binom{j−1}{r} has no hypergeometric closed form, as confirmed by recurrence analysis and the Hyper algorithm.
• The expression W_{n,r} = 2^{n−r}∑_{k=0}^{⌊r/2⌋} binom{n−2−2k}{r−2k} + (−1)^{r+1} is the only one with a fixed number of terms, making it a practical substitute for a closed form.
• The generating function for S_{m,r} is ∑_{m≥2r+2} S_{m,r} x^m = x^{2r+2} / [(1−x)(1−2x)^{r+1}], valid for fixed r ≥ 0.
• The bijection in Lemma 2.6 establishes that |B_{n,r}^{+,o}| = |B_{n,r}^{-,e}| + (−1)^{r+1}, resolving a parity anomaly via an almost-involution with a single fixed point.
• The proof of T_{n,r} = W_{n,r} is completed by showing that the even-weight tilings in B_{n,r}^e contribute ∑_{k=0}^{⌊r/2⌋} 2^{n−r} binom{n−2−2k}{r−2k}, matching the first term of W_{n,r}.