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.

The Springer numbers are defined in connection with the irreducible root system of type $B_n$ and also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of André signed permutations, and by Arnol'd in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length n and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number $B(n,k)$ of labeled ballot paths starting at $(0,0)$ and ending at $(n,k)$. Using our bijection, we find a statistic $\alpha$ such that the number of snakes $\pi$ of type $B_n$ with $\alpha(\pi)=k$ equals $B(n,k)$. We also show that our bijection specializes to a bijection between labeled Dyck paths of length $2n$ and alternating permutations on $[2n]$.