85 research outputs found
Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps
Solving the first nonmonotonic, longer-than-three instance of a classic
enumeration problem, we obtain the generating function of all
1342-avoiding permutations of length as well as an {\em exact} formula for
their number . While achieving this, we bijectively prove that the
number of indecomposable 1342-avoiding permutations of length equals that
of labeled plane trees of a certain type on vertices recently enumerated by
Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number
of rooted bicubic maps enumerated by Tutte in 1963. Moreover, turns out
to be algebraic, proving the first nonmonotonic, longer-than-three instance of
a conjecture of Zeilberger and Noonan. We also prove that
converges to 8, so in particular,
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
We introduce two partially ordered sets, and , of the same
cardinalities as the type-A and type-B noncrossing partition lattices. The
ground sets of and are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which avoid certain
patterns. The order relation is given by (strict) containment of the descent
sets. In each case, by means of an explicit order-preserving bijection, we show
that the poset of restricted permutations is an extension of the refinement
order on noncrossing partitions. Several structural properties of these
permutation posets follow, including self-duality and the strong Sperner
property. We also discuss posets and similarly associated with
noncrossing partitions, defined by means of the excedence sets of suitable
pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
A combinatorial proof of the log-concavity of the numbers of permutations with runs
We combinatorially prove that the number of permutations of length
having runs is a log-concave sequence in , for all . We also give
a new combinatorial proof for the log-concavity of the Eulerian numbers.Comment: 10 pages, 4 figure
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