171 research outputs found
Sorting and preimages of pattern classes
We introduce an algorithm to determine when a sorting operation, such as
stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a
new proof of the description of West-2-stack-sortable permutations, that is
permutations that are completely sorted when passed twice through a stack, in
terms of patterns. We also solve the long-standing problem of describing
West-3-stack-sortable permutations. This requires a new type of generalized
permutation pattern we call a decorated pattern.Comment: 13 pages, 5 figures, to appear at FPSAC 201
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
Motzkin Intervals and Valid Hook Configurations
We define a new natural partial order on Motzkin paths that serves as an
intermediate step between two previously-studied partial orders. We provide a
bijection between valid hook configurations of -avoiding permutations and
intervals in these new posets. We also show that valid hook configurations of
permutations avoiding (or equivalently, ) are counted by the same
numbers that count intervals in the Motzkin-Tamari posets that Fang recently
introduced, and we give an asymptotic formula for these numbers. We then
proceed to enumerate valid hook configurations of permutations avoiding other
collections of patterns. We also provide enumerative conjectures, one of which
links valid hook configurations of -avoiding permutations, intervals in
the new posets we have defined, and certain closed lattice walks with small
steps that are confined to a quarter plane.Comment: 22 pages, 8 figure
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