[Paper Review] A combinatorial proof of strict unimodality for $q$-binomial coefficients

Vivek Dhand|arXiv (Cornell University)|Feb 5, 2014

Advanced Combinatorial Mathematics3 references4 citations

TL;DR

This paper provides a direct combinatorial proof of the strict unimodality of $q$-binomial coefficients ${m+n \choose m}_q$ for $m,n \geq 8$, using O'Hara's structure theorem to decompose the partition lattice $L(m,n)$ into unimodal products of chains. The key contribution is a stronger result: if $m,n \geq 8d$, then the difference between consecutive rank sizes in the middle of the lattice is at least $d$, confirming strict unimodality with explicit quantitative bounds.

ABSTRACT

Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice $L(m,n)$. In fact, we prove a stronger result: if $m, n \geq 8d$, and $2d \leq r \leq mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank.

Motivation & Objective

• To provide a direct combinatorial proof of strict unimodality for $q$-binomial coefficients, avoiding representation-theoretic methods.
• To characterize when products of chains are strictly unimodal, extending O'Hara's structure theorem for the partition lattice $L(m,n)$.
• To establish a quantitative lower bound on the difference $p_r(m,n) - p_{r-1}(m,n)$, showing it is at least $d$ under conditions $m,n \geq 8d$ and $2d \leq r \leq mn/2$.
• To improve upon prior bounds by showing a linear lower bound on $r$ in terms of $d$, as opposed to quadratic or logarithmic estimates.

Proposed method

• Apply a modified version of O'Hara's structure theorem to decompose $L(m,n)$ into ranked subsets $Q_m(d_0,\dots,d_k)$ isomorphic to products of chains.
• Use the characterization of strictly unimodal products of chains via the inequality $a_1 \leq a_2 + \cdots + a_n + 1$ for chain lengths $a_i$.
• Analyze the minimal rank and unimodal structure of $Q_m(d_0,\dots,d_k)$ to derive bounds on rank differences.
• Estimate the number of such unimodal components $N_3(m,n)$ and $N_4(m,n)$ with at least $d$ elements in the desired rank range.
• Use generating function arguments and parity constraints to bound the number of integer solutions to $d_0 + 2d_1 + 3d_2 = n$ under linear constraints on $m$ and $n$.
• Verify that the lowest rank of unimodal components covers the interval $[2d, mn/2]$ when $m,n \geq 8d$, ensuring the difference $p_r - p_{r-1} \geq d$.

Experimental results

Research questions

• RQ1 Can strict unimodality of $q$-binomial coefficients be proven combinatorially without representation theory?
• RQ2 Under what conditions on $m,n,d$ is the difference $p_r(m,n) - p_{r-1}(m,n)$ at least $d$ for $r$ in the upper half of the rank-generating function?
• RQ3 What is the optimal lower bound on $r$ for which strict unimodality with multiplicity $d$ holds, and how does it compare to prior estimates?
• RQ4 How can O'Hara's decomposition into products of chains be used to derive quantitative bounds on rank differences in $L(m,n)$?

Key findings

• For $m,n \geq 8d$ and $2d \leq r \leq mn/2$, the difference $p_r(m,n) - p_{r-1}(m,n) \geq d$, proving strict unimodality with multiplicity $d$.
• The result strengthens the Pak-Panova theorem, which required only $m,n \geq 8$ for $d=1$, by providing a uniform bound for all $d \geq 1$.
• The number of unimodal components $N_3(m,n)$ with three nonzero $d_i$'s is at least $\frac{n-16}{2}$ for $n \geq 24$, ensuring $N_3(m,n) \geq d$ when $d \geq 3$.
• For $d=1$ and $d=2$, the result is verified by checking the existence of at least one or two solutions to Diophantine constraints on $d_0,d_1,d_2,d_3$ for $n \geq 12$ and $n \geq 16$, respectively.
• The lowest rank of the unimodal components $Q_m(1,d_1,d_2,d_3,0,\dots,0)$ is at most $3n-10$, and this is bounded above by $mn/2 - dm$ for $m,n \geq 8d$, ensuring coverage of the interval $[2d, mn/2]$.
• The lower bound on $r$ is linear in $d$, improving upon the quadratic bound in Zanello's earlier work and approaching the optimal logarithmic bound.