51 research outputs found
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
Generating trees and pattern avoidance in alternating permutations
We extend earlier work of the same author to enumerate alternating
permutations avoiding the permutation pattern 2143. We use a generating tree
approach to construct a recursive bijection between the set A_{2n}(2143) of
alternating permutations of length 2n avoiding 2143 and standard Young tableaux
of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating
permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux
of shape (n + 2, n + 1, n). We also give a number of conjectures and open
questions on pattern avoidance in alternating permutations and generalizations
thereof.Comment: 21 pages. To be presented at FPSAC 2010. Comments welcome
Baron Munchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle
We investigate a coin-weighing puzzle that appeared in the Moscow Math
Olympiad in 1991. We generalize the puzzle by varying the number of
participating coins, and deduce an upper bound on the number of weighings
needed to solve the puzzle that is noticeably better than the trivial upper
bound. In particular, we show that logarithmically-many weighings on a balance
suffice.Comment: 19 page
Skyscraper Numbers
We introduce numbers depending on three parameters which we call skyscraper
numbers. We discuss properties of these numbers and their relationship with
Stirling numbers of the first kind, and we also introduce a skyscraper
sequence.Comment: 7 pages, 1 figur
Flashcard games
We study a certain family of discrete dynamical processes introduced by
Novikoff, Kleinberg and Strogatz that we call flashcard games. We prove a
number of results on the evolution of these games, an in particular we settle a
conjecture of NKS on the frequency with which a given card appears. We
introduce a number of generalizations and variations that we believe are of
interest, and provide a large number of open questions and problems.Comment: 13 pages, 2 figure
Pattern avoidance for alternating permutations and reading words of tableaux
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 67-69).We consider a variety of questions related to pattern avoidance in alternating permutations and generalizations thereof. We give bijective enumerations of alternating permutations avoiding patterns of length 3 and 4, of permutations that are the reading words of a "thickened staircase" shape (or equivalently of permutations with descent set {k, 2k, 3k, . . .}) avoiding a monotone pattern, and of the reading words of Young tableaux of any skew shape avoiding any of the patterns 132, 213, 312, or 231. Our bijections include a simple bijection involving binary trees, variations on the Robinson-Schensted-Knuth correspondence, and recursive bijections established via isomorphisms of generating trees.by Joel Brewster Lewis.Ph.D
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