Counting permutations by runs

@article{Zhuang2016CountingPB,
  title={Counting permutations by runs},
  author={Yan Zhuang},
  journal={J. Comb. Theory, Ser. A},
  year={2016},
  volume={142},
  pages={147-176}
}
  • Yan Zhuang
  • Published 9 May 2015
  • Mathematics
  • J. Comb. Theory, Ser. A
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References

SHOWING 1-10 OF 18 REFERENCES
Combinatorics of permutations
TLDR
This book discusses Permutations as Genome Rearrangements, algorithms and permutations, and the proof of the Stanley-Wilf Conjecture.
Decomposition Based Generating Functions for Sequences
Numerous combinatorial enumeration problems may be reduced to equivalent problems of enumerating sequences with prescribed restrictions. For example, the expression, given by Tutte [38], for the
Noncommutative symmetric functions
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an
Introduction to partially ordered patterns
Longest alternating subsequences of permutations
The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group Sn has been the object of much investigation. We develop comparable results for the length as(w) of
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
The On-Line Encyclopedia of Integer Sequences
  • N. Sloane
  • Computer Science
    Electron. J. Comb.
  • 1994
TLDR
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
A coloring problem
Introduction. A well-known algorithm for coloring the vertices of a graph is the "greedy algorithm": given a totally ordered set of colors, each vertex of the graph (taken in some order) is colored
Variations on Descents and Inversions in Permutations
TLDR
The number of alternating inversions is defined in the paper, in alternating (down-up) permutations, and a new $q-analog of the Euler number $E_n$ is obtained and shown how it emerges in a $q$-analogy of an identity expressing $E-n$ as a weighted sum of Dyck paths.
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