Equivalence of extremal conditions in Stanley’s matroid inequality for arbitrary matroids
Prove that for any loopless matroid M of rank n on ground set E and any partition E = R ⊔ Q with both R and Q of rank n, the following conditions are equivalent: (a) for some k ∈ {1,…,n−1}, the log-concavity equality \tilde{N}_k^2 = \tilde{N}_{k−1} \tilde{N}_{k+1} holds; (b) for all k ∈ {1,…,n−1}, \tilde{N}_k^2 = \tilde{N}_{k−1} \tilde{N}_{k+1} holds; and (c) there exist positive integers r, q such that |\overline{x} ∩ R|·r = |\overline{x} ∩ Q|·q for every x ∈ E, where \overline{x} denotes the parallel class of x. Here N_k is the number of bases of M with exactly k elements in R and \tilde{N}_k = N_k / \binom{n}{k}.
References
Thus, we make the following conjecture for all matroids (not necessarily regular).
Conjecture \ref{matroid-conjecture} Let $M$ be a loopless matroid of rank $n$ on a set $E$. Let $R \subseteq E$ be a subset and let $Q = E \backslash R$. For $k$, $0 \leq k \leq n$, we define $N_k$ to be the number of bases of $M$ with $k$ elements in $R$. Define $\widetilde{N}k = \frac{N_k}{\binom{n}{k}}$. Suppose that $R$ and $Q$ both have rank $n$. Then, the following are equivalent. (a) $\widetilde{N}_k2 = \widetilde{N}{k-1} \widetilde{N}{k+1}$ for some $k \in {1, \ldots, n-1}$. (b) $\widetilde{N}_k2 = \widetilde{N}{k-1} \widetilde{N}_{k+1}$ for all $k \in {1, \ldots, n-1}$. (c) There are positive integers $r, q \geq 1$ such that $|\overline{x} \cap R| r = |\overline{x} \cap Q| q$ for all $x \in E$.