An odd diagram class is a set of permutations with the same odd diagram. Brenti, Carnevale, and Tenner [ Comb. Theory, 2 (2022), 13] showed that each odd diagram class is an interval in the Bruhat order. They conjectured that such intervals are rank-symmetric. In this paper, we present an algorithm to partition an odd diagram class in a uniform manner. As an application, we obtain that the Poincaré polynomial of an odd diagram class factors into polynomials of the form $1+t+\cdots+t^m$. This, in particular, resolves the conjecture of Brenti, Carnevale, and Tenner [ Comb. Theory, 2 (2022), 13].

We confirm a conjecture of Monical, Tokcan, and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.

Barely set-valued tableaux were introduced by Reiner, Tenner, and Yong in their study of the probability distribution of edges in the Young lattice of partitions. We prove a generalization of a conjecture of Reiner, Tenner, and Yong on the number of barely set-valued tableaux. To do this we apply results of Chan, Haddadan, Hopkins, and Moci on jaggedness of shapes.

Given a sequence $s=(s_1,s_2,\ldots )$ of positive integers, the notion of inversion sequences with respect to $s$, or $s$-inversion sequences, was introduced by Savage and Schuster in their study of lecture hall polytopes. A sequence $(e_1,e_2,\ldots,e_n)$ of nonnegative integers is called an $s$-inversion sequence of length $n$ if $0\leq e_i < s_i$ for $1\leq i\leq n$. Let $I_n$ be the set of $s$-inversion sequences of length $n$ for $s=(1,4,3,8,5,12,\ldots)$---that is, $s_{2i-1}=2i-1$ and $s_{2i}=4i$ for $i\geq1$---and let $P_n$ be the set of signed permutations on the multiset $\{1^2,2^2,\ldots,n^2\}$. Savage and Visontai conjectured that the descent number over $P_n$ is equidistributed with the ascent number over $I_{2n}$. In this paper, we give a proof of this conjecture by using $P$-partitions of type $B$. (Lin independently obtained a proof based on recurrence relations.) Moreover, we find a set of signed permutations over which the descent number is equidistributed with the ascent number over $I_{2n-1}$. Let $I'_n$ be the set of $s$-inversion sequences of length $n$ for $s=(2,2,6,4,10,6,\ldots)$; that is, $s_{2i-1}=4i-2$ and $s_{2i}=2i$ for $i\geq1$. We also find two sets of signed permutations over which the descent number is equidistributed with the ascent number over $I'_n$, depending on the parity of $n$.