Tableau sequences, open diagrams, and Baxter families

Abstract

Abstract. Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -open diagrams. Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case.

Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods

“…It is convenient to not represent the null step in the drawings, but to group the steps by pairs of the form (s 2i−1 , s 2i ) instead. In Section 4 we show the analogous, although more difficult, result for hesitating walks, which answers a recent question of Burrill et al [8]:…”

“…Hesitating excursions of half-length n − 1 in the quadrant are known to be counted by the Baxter numbers B n = 2 n(n+1) 2 n k=1 n+1 k+1 n+1 k n+1 k−1 . Indeed, as shown by Burrill et al [8], they are in easy bijection with the classical Baxter family of non-intersecting triples of directed lattice walks. On the other hand it has been first shown in [31] (and more recently in [8]) that hesitating axis-walks of half-length n in the octant are also counted by B n+1 .…”

On Standard Young Tableaux of Bounded Height

“…We have outlined one possible argument, but in fact three very different proofs have been discovered. Above is the argument of Burrill, Courtiel, Fusy, Melczer and Mishna [9]. Krattenthaler [28] determines a different bijection using growth diagrams as an intermediary object.…”

Polynomiality of Certain Average Weights for Oscillating Tableaux

“…Another proof was obtained by Stanley [25,27] in the study of differential posets. The enumerations of various oscillating tableaux with restrictive conditions can be found in [3,4,6,17,16,22]. In 2015, Hopkins and Zhang [13] proved the following result on the average of certain weight function of oscillating tableaux.…”