Abstract. We present some conjectures and open problems on partition hook lengths, which are all motivated by known results on the subject. The conjectures are suggested by extensive experimental calculations using a computer algebra system. The first conjecture unifies two classical results on the number of standard Young tableaux and the number of pairs of standard Young tableaux of the same ...

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 an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functio...

Oscillating tableaux are certain walks in Young’s lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all t...

Stanley conjectured that the number of maximal chains in the weak Bruhat order of Sn, or equivalently the number of reduced decompositions of the reverse of the identity permutation w0 = n, n− 1, n − 2, . . . , 2, 1, equals the number of standard Young tableaux of staircase shape s = {n− 1, n− 2, . . . , 1}. Originating from this conjecture remarkable connections between standard Young tableaux...

A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equidistributed with the standard one. This concept then applied construct explicit cyclic extensions Young tableaux Motzkin paths. Schur-positivity of associated quasisymmetric functions follows.

The conjugacy classes of nilpotent n × n matrices can be parametrised by partitions λ of n, and for a nilpotent η in the class parametrised by λ, the variety Fη of η-stable flags has its irreducible components parametrised by the standard Young tableaux of shape λ. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Stei...

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let M be a finite dimensional representation of a semi-simple Lie algebra and let B be the associated Kashiwara crystal. For r > 0, the triple (X, c, P ) which exhibits the cyclic sieving phenomenon is constructed as follows: the set X is the set of isolat...

As is well known, Young diagrams consisting of n boxes parameterize isomorphism classes of finite-dimensional irreducible representations of the symmetric group Sn of degree n, and moreover the structure of each irreducible representation is described in terms of tableaux on the corresponding Young diagram; namely, a basis of the representation is labeled by standard tableaux, with which the ac...

This paper contains a short proof of a formula by Frame, Robinson, and Thrall [I] h h w ic counts the number of Young tableaux of a given shape. Let X = {X, >, X, > ..’ 2 X,} be a partition of R. The Ferrers diagram of h is an array of cells doubly indexed by pairs (i, j) with 1 < i < m, 1 <j < A( . A Young tableau of shape h (sometimes called a standard tableau) is an arrangement of the intege...