Houyi Yu

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the hyperoctahedral group $B_n$. This bijection preserves descent statistics and induces a poset isomorphism under the left weak order. As a consequence, separable signed permutations are enumerated by the large Schröder numbers, and their descent polynomials are shown to be $γ$-positive. Building on a recursive characterization of separable signed permutations via direct sum and skew sum operations, we derive explicit product formulas for the rank generating functions of the principal upper and lower ideals of separable signed permutations under the left weak order.

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type $A$ to all Coxeter types. In this paper, we give an equivalent description, in a type-independent manner, when two left weak Bruhat intervals in a Coxeter group are descent-preserving isomorphic. As an application, we classify all left weak Bruhat interval modules of $0$-Hecke algebras up to isomorphism, and thereby answer an open question and resolve in the affirmative a conjecture of Jung, Kim, Lee, and Oh. Additionally, for finite Coxeter groups we show that the set of minimum (or maximum) elements of all left weak Bruhat intervals in each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.

For the Malvenuto-Reutenauer Hopf algebra of permutations, we provide a cancellation-free antipode formula for any permutation of the form $ab1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, which starts with the decreasing sequence $ab$ and ends with the increasing sequence $1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, where $1\leq b<a\leq n$. As a consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce E. Sagan.

Given a positive integer $n$ and a nonnegative integer $k$ with $k\leq n$, we denote by $\mathcal{A}(n,k)$ the class of all $n$-by-$n$ $(0,1)$-matrices with constant row and column sums $k$. In this paper, we show that the Bruhat order and the secondary Bruhat order coincide on $\mathcal{A}(n,k)$ if and only if either $0\leq n\leq 5$ or $k\in\{0,1,2,n-2,n-1,n\}$ with $n\geq 6$.