429 research outputs found
Enumeration of a dual set of Stirling permutations by their alternating runs
In this paper, we count a dual set of Stirling permutations by the number of
alternating runs. Properties of the generating functions, including recurrence
relations, grammatical interpretations and convolution formulas are studied.Comment: 8 page
Counting permutations by alternating descents
We find the exponential generating function for permutations with all valleys
even and all peaks odd, and use it to determine the asymptotics for its
coefficients, answering a question posed by Liviu Nicolaescu. The generating
function can be expressed as the reciprocal of a sum involving Euler numbers.
We give two proofs of the formula. The first uses a system of differential
equations. The second proof derives the generating function directly from
general permutation enumeration techniques, using noncommutative symmetric
functions. The generating function is an "alternating" analogue of David and
Barton's generating function for permutations with no increasing runs of length
3 or more. Our general results give further alternating analogues of
permutation enumeration formulas, including results of Chebikin and Remmel
Introduction to Partially Ordered Patterns
We review selected known results on partially ordered patterns (POPs) that
include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified)
maxima and minima) in permutations, the Horse permutations and others. We
provide several (new) results on a class of POPs built on an arbitrary flat
poset, obtaining, as corollaries, the bivariate generating function for the
distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and
Pell numbers, as well as generalizations of few results in the literature
including the descent distribution. Moreover, we discuss q-analogue for a
result on non-overlapping segmented POPs. Finally, we suggest several open
problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea
Some combinatorial arrays related to the Lotka-Volterra system
The purpose of this paper is to investigate the connection between the
Lotka-Volterra system and combinatorics. We study several context-free grammars
associated with the Lotka-Volterra system. Some combinatorial arrays, involving
the Stirling numbers of the second kind and Eulerian numbers, are generated by
these context-free grammars. In particular, we present grammatical
characterization of some statistics on cyclically ordered partitions.Comment: 15 page
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