Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

A maxdrop statistic for standard Young tableaux

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.

Skew characters and cyclic sieving

In 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of the cyclic sieving phenomenon. We extend this result to m-tuples of skew standard Young tableaux of the same shape, for fixed m, subject to the condition that the mth power of the associated fake-degree polynomial evaluates to nonnegative integers at roots of unity. However, we are unable to specify an explicit group action. Put differently, we determine in which cases the mth tensor power of a skew character of the symmetric group carries a permutation representation of the cyclic group. To do so, we use a method proposed by Amini and the first author, which amounts to establishing a bound on the number of border-strip tableaux of skew shape. Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a bijection between permutations and Stembridge’s alternating tableaux, which intertwines rotation and promotion.

Commutation Classes of the Reduced Words for the Longest Element of ${\mathfrak S}_n$

Using the standard Coxeter presentation for the symmetric group $\mathfrak{S}_{n}$, two reduced expressions for the same group element $\textsf{w}$ are said to be commutationally equivalent if one expression can be obtained from the other one by applying a finite sequence of commutations. The commutation classes can be seen as the vertices of a graph $\widehat{G}(\textsf{w})$, where two classes are connected by an edge if elements of those classes differ by a long braid relation. We compute the radius and diameter of the graph $\widehat{G}(\textsf{w}_{\bf 0})$, for the longest element $\textsf{w}_{\bf 0}$ in the symmetric group $\mathfrak{S}_{n}$, and show that it is not a planar graph for $n\geq 6$. We also describe a family of commutation classes which contains all atoms, that is classes with one single element, and a subfamily of commutation classes whose elements are in bijection with standard Young tableaux of certain moon-polyomino shapes.

Hook formulas for skew shapes

International audience The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.

On Cyclic Schur-Positive Sets of Permutations

We introduce a notion of cyclic Schur-positivity for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures by the last two authors and by Adin–Gessel–Reiner–Roichman, and yields new examples of Schur-positive sets.