Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Abstract

We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

Diagram supermodules for $0$-Hecke-Clifford algebras

“…Demonstrating the wide applicability of the diagram modules framework, we show that the 0-Hecke modules constructed for notable bases of QSym in each of [4,6,31,34,38] are cases of diagram 0-Hecke modules. Extending this further, we show that the 0-Hecke modules constructed from permuted tableaux in [12,23,39], and the projective indecomposable 0-Hecke modules constructed in terms of ribbon tableaux in [21], are likewise obtained as diagram 0-Hecke modules. Diagram 0-Hecke modules moreover generalize the family of weak Bruhat interval modules considered by Jung, Kim, Lee and Oh [23] as an alternative framework for studying the 0-Hecke modules in [4,6,34,38,39].…”

“…Extending this further, we show that the 0-Hecke modules constructed from permuted tableaux in [12,23,39], and the projective indecomposable 0-Hecke modules constructed in terms of ribbon tableaux in [21], are likewise obtained as diagram 0-Hecke modules. Diagram 0-Hecke modules moreover generalize the family of weak Bruhat interval modules considered by Jung, Kim, Lee and Oh [23] as an alternative framework for studying the 0-Hecke modules in [4,6,34,38,39]. We prove that every weak Bruhat interval module can be realized as a diagram 0-Hecke module; on the other hand, there exist diagram 0-Hecke modules that are not isomorphic to any weak Bruhat interval module.…”

“…There has been much recent activity concerning construction of 0-Hecke modules whose quasisymmetric characteristics are elements of noteworthy bases of QSym, e.g., [4,6,31,34,38,39], and also in determining the structure of these modules, e.g., [12,13,14,23,25]. It is therefore natural to ask the same question for 0-Hecke-Clifford algebras: can one construct 0-Hecke-Clifford supermodules whose peak characteristics are noteworthy families or bases of functions in Peak?…”

Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

“…(b) Besides dual immaculate functions, the problem of constructing H n (0)-modules has been considered for the following quasisymmetric functions: the quasisymmetric Schur functions in [27,28], the extended Schur functions in [26], the Young row-strict quasisymmetric Schur functions in [2], the Young quasisymmetric Schur functions in [12], and the images of all these quasisymmetric functions under certain involutions on QSym in [21]. Although these modules are built in a very similar way, their homological properties have not been well studied.…”

Smirnov Trees

“…Gessel [6] was the first to study the analogue of the descent statistic for labeled binary trees, and he further pointed out intriguing connections to the enumerative theory of Coxeter arrangements. There has been a flurry of activity towards understanding these connections better, and the reader is referred to [2,3,4,7,19,20] for more details. Gessel-Griffin-Tewari [7] investigated these connections from the perspective of symmetric functions; they attach a multivariate generating function tracking ascent-descents over all labeled binary trees and subsequently prove that this generating function expands positively in terms of ribbon Schur functions.…”