The k-Schur functions were first introduced by Lapointe, Lascoux and Morse [18] in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono [17] as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Br...

We present a set of seminar notes on basic concepts and results on automated and semi-automated veriication of correctness and validation of software systems (protocols) via the method of tableaux, often called semantic tableaux. Special emphasis is on comparison of various ways of formalization of the concept of a `proof', on the example of the classical rst-order logic.

The cactus group acts on the set of standard Young tableaux a given shape by (partial) Schützenberger involutions. It is natural to extend this action corresponding Specht module identifying with Kazhdan–Lusztig basis. We term these representations “Schützenberger modules”, denoted S Sch λ , and in paper we investigate their decomposition into irreducible components. prove that when hook shape,...

We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper Re], we start with probability distributions on the space of the innnite Young tableaux. Then we calculate the probability that the entry of a random tableau at a given box equals n = 1; 2; : : :. Summing these probabilities over n and equa...

Let X, Y be finite sets and T a set of functions from X → Y which we will call “tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistan...

We define P-strict labelings for a finite poset P as generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with toggle action B-bounded Q-partitions an associated Q. In many nice cases, this conjugate to rowmotion. apply result flagged tableaux, Gelfand-Tsetlin patterns, symplectic obtaining new cyclic sieving homomesy conjectures. al...

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit q = t = 0 and q = t→∞, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition fo...

We compute the dimension of some irreducible representations of the sym­ metric groups in characteristic ρ (Theorem 2). The representations considered here are associated with Young diagrams m : πΐχ > m 2 > ... > m/ such that πΐχ — mi < (jp — I). The formula is based on a variant of Verlinde's formula which computes some tensor product multiplicities of indecomposable modules for GLi(¥p), as it...

We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical...

We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to recover a formula for Eulerian numbers which is a direc...