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Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 106 (8 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n 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, H(x) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
A Survey of StackSorting Disciplines
, 2004
"... We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as You ..."
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We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as Young tableau, planar graph theory, and simplicial complexes.
2Stack Sortable Permutations With A Given Number Of Runs
, 1997
"... . Using earlier results we prove a formula for the number W (n;k) of 2stack sortable permutations of length n with k runs, or in other words, k \Gamma 1 descents. This formula will yield the suprising fact that there are as many 2stack sortable permutations with k \Gamma 1 descents as with k \Gamma ..."
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. Using earlier results we prove a formula for the number W (n;k) of 2stack sortable permutations of length n with k runs, or in other words, k \Gamma 1 descents. This formula will yield the suprising fact that there are as many 2stack sortable permutations with k \Gamma 1 descents as with k \Gamma 1 ascents. We also prove that W (n;k) is unimodal in k, for any fixed n. 1. Introduction 1.1. Our main results. In this paper we are going to show that the number of 2stack sortable permutations of length n with k \Gamma 1 ascents is equal to the number T (n;k) of fi(1; 0)trees on n + 1 nodes with k leaves. (See Section 2 for the definition of fi(1; 0)trees). This, and results from [2] and [5] will enable us to easily show that W (n;k) = (n + k \Gamma 1)!(2n \Gamma k)! k!(n + 1 \Gamma k)!(2k \Gamma 1)!(2n \Gamma 2k + 1)! ; (1) which formula is symmetric in k and n + 1 \Gamma k. 1.2. Background and Definitions. In what follows, permutations of length n will be called npermutations...
Workshop permutation patterns 2005 List of Abstracts
, 2005
"... The asterisk denotes the abstract sender, presumably also the speaker. The list is in alphabetical order by the first author. 1. Dan Bernstein ∗ and Amitai Regev. A Foata bijection for the alternating group and for qanalogues. ..."
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The asterisk denotes the abstract sender, presumably also the speaker. The list is in alphabetical order by the first author. 1. Dan Bernstein ∗ and Amitai Regev. A Foata bijection for the alternating group and for qanalogues.
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"... 2009 To my family i ii ACKNOWLEDGMENTS Many people have played a role either directly or indirectly in shaping this thesis. I am indebted to my advisors Cristian Estan and Somesh Jha, who over the course of the last few years guided me throughout this work. Their insights, energy, and encouragement ..."
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2009 To my family i ii ACKNOWLEDGMENTS Many people have played a role either directly or indirectly in shaping this thesis. I am indebted to my advisors Cristian Estan and Somesh Jha, who over the course of the last few years guided me throughout this work. Their insights, energy, and encouragement where invaluable to me. I have had the privilege of collaborating with many great students on problems directly related to this work as well as problems that were more peripheral. I would like to express my thanks to Shijin Kong, Dan Gibson, Daniel Luchaup, Drew Davidson, and Lorenzo De Carli for the opportunity to work with them. My parents, Johnson and Rhea Smith, have always provided encouragement and support, even when they could not understand and I could not explain some of the challenges of the moment. Over the past few years, my sister Tiffany Morgan always provided invaluable help, even when I was perhaps undeserving. Thank you. Mom, Dad, and Tiffany: you have helped to make this dissertation possible.
2STACK SORTABLE PERMUTATIONS WITH A GIVEN NUMBER OF RUNS
, 1997
"... Abstract. Using earlier results we prove a formula for the number W (n,k) of 2stack sortable permutations of length n with k runs, or in other words, k − 1 descents. This formula will yield the suprising fact that there are as many 2stack sortable permutations with k −1 descents as with k −1 ascen ..."
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Abstract. Using earlier results we prove a formula for the number W (n,k) of 2stack sortable permutations of length n with k runs, or in other words, k − 1 descents. This formula will yield the suprising fact that there are as many 2stack sortable permutations with k −1 descents as with k −1 ascents. We also prove that W (n,k) is unimodal in k, for any fixed n. 1.