Yuping Yang

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.

Let P be a partially ordered set, $\mathcal{R}$ a 2-torsion free commutative ring with unity and $FI(P,\mathcal{R})$ the finitary incidence algebra of P over $\mathcal{R}$. In this paper, we give an explicit description for the structure of nonlinear Jordan derivations of $FI(P,\mathcal{R})$. We show that if P has no trivial component, then every nonlinear Jordan derivation of $FI(P,\mathcal{R})$ is proper, and can be presented as a sum of a generalized inner derivation, a transitive induced derivation and an additive induced derivation. If P has a trivial connected component, we prove that every nonlinear Jordan derivation of $FI(P,\mathcal{R})$ is proper if and only if every nonlinear Jordan derivation of R is also proper.

Let $(X,\le )$ be a locally finite preordered set, $\mathcal{R}$ a two-torsion-free commutative ring with unity and $I(X,\mathcal{R})$ the incidence algebra of X over $\mathcal{R}.$ In this paper, all the nonlinear Jordan derivations of $I(X,\mathcal{R})$ are determined. In particular, if all the connected components of X are nontrivial, we prove that every nonlinear Jordan derivation of $I(X,\mathcal{R})$ is proper and can be presented as a sum of an inner derivation, a transitive induced derivation, and an additive induced derivation.

Let n be a positive integer with $n\ge 2.$ Let X be a locally finite preordered set, R a commutative ring with unity and I(X, R) the incidence algebra of X over R. We prove in this article that every Lie n-derivation of I(X, R) is proper provided that R is 2-torsion free and $(n-1)$-torsion free, which gives a positive answer for a conjecture by Wang and Xiao. see [Lie triple derivations of incidence algebras. Commun. Algebra, 47(5): 1841-1852.]

Let $(X,\le )$ be a finite preordered set, R a 2-torsion free commutative ring with unity and $I(X,R)$ the incidence algebra of X over R. In this paper we prove that every nonlinear Lie derivation of $I(X,R)$ is of the standard form. More explicitly, each nonlinear Lie derivation of $I(X,R)$ is a sum of an inner derivation, a transitive induced derivation, an additive induced derivation and a central-valued map.