975 research outputs found
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
We prove that the Stanley-Wilf limit of any layered permutation pattern of
length is at most , and that the Stanley-Wilf limit of the
pattern 1324 is at most 16. These bounds follow from a more general result
showing that a permutation avoiding a pattern of a special form is a merge of
two permutations, each of which avoids a smaller pattern. If the conjecture is
true that the maximum Stanley-Wilf limit for patterns of length is
attained by a layered pattern then this implies an upper bound of for
the Stanley-Wilf limit of any pattern of length .
We also conjecture that, for any , the set of 1324-avoiding
permutations with inversions contains at least as many permutations of
length as those of length . We show that if this is true then the
Stanley-Wilf limit for 1324 is at most
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,
Real Zeros and Partitions without singleton blocks
We prove that the generating polynomials of partitions of an -element set
into non-singleton blocks, counted by the number of blocks, have real roots
only and we study the asymptotic behavior of the leftmost roots. We apply this
information to find the most likely number of blocks.Comment: 16 page
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
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