10,760 research outputs found
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
Old and young leaves on plane trees
A leaf of a plane tree is called an old leaf if it is the leftmost child of
its parent, and it is called a young leaf otherwise. In this paper we enumerate
plane trees with a given number of old leaves and young leaves. The formula is
obtained combinatorially by presenting two bijections between plane trees and
2-Motzkin paths which map young leaves to red horizontal steps, and old leaves
to up steps plus one. We derive some implications to the enumeration of
restricted permutations with respect to certain statistics such as pairs of
consecutive deficiencies, double descents, and ascending runs. Finally, our
main bijection is applied to obtain refinements of two identities of Coker,
involving refined Narayana numbers and the Catalan numbers.Comment: 11 pages, 7 figure
The structure of the consecutive pattern poset
The consecutive pattern poset is the infinite partially ordered set of all
permutations where if has a subsequence of adjacent
entries in the same relative order as the entries of . We study the
structure of the intervals in this poset from topological, poset-theoretic, and
enumerative perspectives. In particular, we prove that all intervals are
rank-unimodal and strongly Sperner, and we characterize disconnected and
shellable intervals. We also show that most intervals are not shellable and
have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR
Two permutation classes enumerated by the central binomial coefficients
We define a map between the set of permutations that avoid either the four
patterns or , and the set of Dyck
prefixes. This map, when restricted to either of the two classes, turns out to
be a bijection that allows us to determine some notable features of these
permutations, such as the distribution of the statistics "number of ascents",
"number of left-to-right maxima", "first element", and "position of the maximum
element"Comment: 26 pages, 3 figure
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