172 research outputs found
Enumeration of Stack-Sorting Preimages via a Decomposition Lemma
We give three applications of a recently-proven "Decomposition Lemma," which
allows one to count preimages of certain sets of permutations under West's
stack-sorting map . We first enumerate the permutation class
, finding a new example
of an unbalanced Wilf equivalence. This result is equivalent to the enumeration
of permutations sortable by , where is the bubble
sort map. We then prove that the sets ,
,
and are
counted by the so-called "Boolean-Catalan numbers," settling a conjecture of
the current author and another conjecture of Hossain. This completes the
enumerations of all sets of the form
for
with the exception of the set
. We also find an explicit formula for
, where
is the set of permutations in with descents.
This allows us to prove a conjectured identity involving Catalan numbers and
order ideals in Young's lattice.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1903.0913
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
Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
We study a two-parameter generalization of the Catalan numbers:
is the number of ways to subdivide the -dimensional hypercube into
rectangular blocks using orthogonal partitions of fixed arity . Bremner \&
Dotsenko introduced in their work on Boardman--Vogt tensor
products of operads; they used homological algebra to prove a recursive formula
and a functional equation. We express as simple finite sums, and
determine their growth rate and asymptotic behaviour. We give an elementary
proof of the functional equation, using a bijection between hypercube
decompositions and a family of full -ary trees. Our results generalize the
well-known correspondence between Catalan numbers and full binary trees
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