27 research outputs found
Set-Valued Tableaux & Generalized Catalan Numbers
Standard set-valued Young tableaux are a generalization of standard Young
tableaux in which cells may contain more than one integer, with the added
conditions that every integer at position must be smaller than every
integer at positions and . This paper explores the
combinatorics of standard set-valued Young tableaux with two-rows, and how
those tableaux may be used to provide new combinatorial interpretations of
generalized Catalan numbers. New combinatorial interpretations are provided for
the two-parameter Fuss-Catalan numbers (Raney numbers), the rational Catalan
numbers, and the solution to the so-called "generalized tennis ball problem".
Methodologies are then introduced for the enumeration of standard set-valued
Young tableaux, prompting explicit formulas for the general two-row case. The
paper closes by drawing a bijection between arbitrary classes of two-row
standard set-valued Young tableaux and collections of two-dimensional lattice
paths that lie weakly below a unique maximal path
On the -Enumeration of Barely Set-Valued Tableaux and Plane Partitions
Barely set-valued tableaux are a variant of Young tableaux in which one box
contains two numbers as its entry. It has recently been discovered that there
are product formulas enumerating certain classes of barely set-valued tableaux.
We give some q-analogs of these product formulas by introducing a version of
major index for these tableaux. We also give product formulas and q-analogs for
barely set-valued plane partitions. The proofs use several probability
distributions on the set of order ideals of a poset, depending on the real
parameter q > 0, which we think could be of independent interest.Comment: 38 pages, 6 tables, 3 figure