171 research outputs found
Staircases, dominoes, and the growth rate of 1324-avoiders
We establish a lower bound of 10.271 for the growth rate of the permutations avoiding 1324, and an upper bound of 13.5. This is done by first finding the precise growth rate of a subclass whose enumeration is related to West-2-stack-sortable permutations, and then combining copies of this subclass in particular ways
A Distribution Function Arising in Computational Biology
Karlin and Altschul in their statistical analysis for multiple high-scoring
segments in molecular sequences introduced a distribution function which gives
the probability there are at least r distinct and consistently ordered segment
pairs all with score at least x. For long sequences this distribution can be
expressed in terms of the distribution of the length of the longest increasing
subsequence in a random permutation. Within the past few years, this last
quantity has been extensively studied in the mathematics literature. The
purpose of these notes is to summarize these new mathematical developments in a
form suitable for use in computational biology.Comment: 9 pages, no figures. Revised version makes minor change
Combinatorial properties of the numbers of tableaux of bounded height
We introduce an infinite family of lower triangular matrices ¡(s), where
°s
n;i counts the standard Young tableaux on n cells and with at most
s columns on a suitable subset of shapes. We show that the entries
of these matrices satisfy a three-term row recurrence and we deduce
recursive and asymptotic properties for the total number ¿s(n) of
tableaux on n cells and with at most s columns
Combinatorial properties of the numbers of tableaux of bounded height
We introduce an infinite family of lower triangular matrices ¡(s), where
°s
n;i counts the standard Young tableaux on n cells and with at most
s columns on a suitable subset of shapes. We show that the entries
of these matrices satisfy a three-term row recurrence and we deduce
recursive and asymptotic properties for the total number ¿s(n) of
tableaux on n cells and with at most s columns
Kronecker product identities from D-finite symmetric functions
Using an algorithm for computing the symmetric function Kronecker product of
D-finite symmetric functions we find some new Kronecker product identities. The
identities give closed form formulas for trace-like values of the Kronecker
product.Comment: 6 page
Alternating, pattern-avoiding permutations
We study the problem of counting alternating permutations avoiding
collections of permutation patterns including 132. We construct a bijection
between the set S_n(132) of 132-avoiding permutations and the set A_{2n +
1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k
of patterns and certain related patterns q_1, ..., q_k, our bijection restricts
to a bijection between S_n(132, p_1, ..., p_k), the set of permutations
avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of
alternating permutations avoiding 132 and the q_i. This reduces the enumeration
of the latter set to that of the former.Comment: 7 page
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