Ascent sequences were introduced by the author (in conjunction with others) to encode a class of permutations that avoid a single length- three bivincular pattern, and were the central object through which other combinatorial correspondences were discovered. In this note we prove the non-trivial fact that generalized ballot sequences are ascent sequences

Dukes, Mark

Abstract. Ascent sequences were introduced by the author (in conjunction with others) to encode a class of permutations that avoid a single length-three bivincular pattern, and were the central object through which other combinatorial correspondences were discovered. In this note we prove the non-trivial fact that generalized ballot sequences are ascent sequences. Ascent sequences were introduced in [1] to encode a collection of permutations that avoid the single length-three bivincular pattern 2|31. These sequences were also shown to uniquely encode interval orders, Stoimenow matchings, and the set of upper triangular matrices whose entries are non-negative integers and which contain no all-zero rows or columns [2]. Generalized ballot sequences (also known as Yamanouchi words) are sequences of non-negative integers that encode election scenarios in which a prescribed set of candidates maintain their success-positions throughout the counting of the votes (i.e., reading of the ballot sequences from left to right). It is not immediately clear that generalized ballot sequences satisfy the defining property of ascent sequence

Mark Dukes

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GENERALIZED BALLOT SEQUENCES ARE ASCENT SEQUENCES

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Strathprints Institutional RepositoryDukes, Mark (2016) Generalized ballot sequences are ascent sequences. Australasian Journal of Combinatorics, 64 (1). pp. 61-63. ISSN 1034-4942 , This version is available at http://strathprints.strath.ac.uk/54428/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukGENERALIZED BALLOT SEQUENCES ARE ASCENTSEQUENCESMARK DUKESDepartment of Computer and Information Sciences,University of Strathclyde,Glasgow, U.K.Abstract. Ascent sequences were introduced by the author (in conjunctionwith others) to encode a class of permutations that avoid a single length-three bivincular pattern, and were the central object through which othercombinatorial correspondences were discovered. In this note we prove thenon-trivial fact that generalized ballot sequences are ascent sequences.Ascent sequences were introduced in [1] to encode a collection of permutationsthat avoid the single length-three bivincular pattern 2|31. These sequences werealso shown to uniquely encode interval orders, Stoimenow matchings, and the setof upper triangular matrices whose entries are non-negative integers and whichcontain no all-zero rows or columns [2].Generalized ballot sequences (also known as Yamanouchi words) are sequencesof non-negative integers that encode election scenarios in which a prescribed set ofcandidates maintain their success-positions throughout the counting of the votes(i.e., reading of the ballot sequences from left to right). It is not immediately clearthat generalized ballot sequences satisfy the defining property of ascent sequencesand this is the purpose of this note.Given a sequence of integers x = (x1, . . . , xn) we say there is an ascent at positioni if xi < xi+1, and denote by asc(x) the number of ascents in x. We say that asequence of positive integers x = (x1, . . . , xn) is an ascent sequence if x1 = 1 andfor all 1 ≤ i < n we havexi+1 ∈ {1, 2, . . . , 2 + asc(x1, . . . , xi)}.This definition is equivalent to the definition given in [2] – the only difference beingthat unity has been added to every sequence entry. Let Ascn be the set of ascentsequences of length n. For exampleAsc3 = {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (1, 2, 3)}.Let w = w1w2 . . . wn be a word over the alphabet {1, 2, . . . , n}. Let us writeasc(w) for asc(w1, . . . , wn). The word w is said to be a generalized ballot sequenceor a Yamanouchi word (see e.g. Stanley [4, Prop. 7.10.3]) if for every left factorw(k) = w1 . . . wk of w and for every i, the number of occurrences of i in w(k) isgreater than or equal to the number of occurrences of i + 1 in w(k). Let Ballotnbe the set of generalized ballot sequences of length n. For example, Ballot3 ={111, 112, 121, 123}.12 GENERALIZED BALLOT SEQUENCES ARE ASCENT SEQUENCESTheorem 1. If x1x2 . . . xn is a generalized ballot sequence, then (x1, . . . , xn) is anascent sequence.Proof. Given x = x1 . . . xn ∈ Ballotn, let max(x) = max{x1, . . . , xn},c(i)j = |{k : xk = j and k ≤ i}|, andValid(x) = {1} ∪ {j : c(n)j−1 > c(n)j and 1 < j ≤ max(x)} ∪ {max(x) + 1}.The set Valid(x) is the set of values that may be appended to x in order to yield ageneralized ballot sequence of length n+ 1. We give a proof by induction.The theorem is true for n = 1 since Ballot1 = {1} and Asc1 = {(1)}. Suppose theclaim to be true for n = m. Let x = x1 . . . xm ∈ Ballotm. Since x is a generalizedballot sequence, all of the letters from {1, . . . ,max(x)} appear at least once in x.Furthermore, the leftmost (i.e., first) occurrence of k in x means that the letterimmediately to its left must be less than it (for otherwise the generalized ballotsequence property would be broken for the left factor of x that ends at the entryimmediately preceding k). Let ak be the smallest index j such that xj = k. Theabove reasoning means we must have xak−1 < xak for all k = 2, . . . ,max(x), inother words (x1, . . . , xm) has ascents at positions a2 − 1, . . . , amax(x) − 1. Thisimplies there are at least max(x)− 1 different ascents in x, i.e.,max(x1, . . . , xm)− 1 ≤ asc(x1, . . . , xm). (1)Every x′ = x1 . . . xmxm+1 ∈ Ballotm+1 is uniquely formed from x = x1 . . . xm ∈Ballotm and xm+1 ∈ Valid(x). From the induction hypothesis, (x1, . . . , xm) ∈ Ascm.Since xm+1 ∈ Valid(x), we have xm+1 ∈ {1, . . . ,max(x) + 1}. Using the inequalityin (1), since max(x) ≤ 1 + asc(x) we havexm+1 ∈ {1, . . . ,max(x) + 1} ⊆ {1, . . . , 2 + asc(x)}.This condition implies x′ = (x1, . . . , xm+1) ∈ Ascm+1. Therefore, by the principleof induction, the claim is true for all positive integers n. Ballot sequences are word-encodings of standard Young tableaux. We posit thatemulating operations on standard Young tableaux, such as evacuation and flippingabout the diagonal, will have some part to play in answering unsolved questionsregarding ascent sequences. One such open problem is describing the ‘dual’ or ‘flip’of an ascent sequence (see [2, Question 20]).An intriguing problem is to classify the subsets of the four classes of combinatorialobjects (pattern-avoiding permutations, Stoimenow matchings, restricted integermatrices, interval orders) that correspond to ballot sequences via the bijectionsgiven in the papers [1, 2, 3].References[1] M. Bousquet-Me´lou, A. Claesson, M. Dukes and S. Kitaev. (2+2)-free posets, ascent sequencesand pattern avoiding permutations. J. Combin. Theory Ser. A 117 (2010), no. 7, 884–909.[2] M. Dukes and R. Parviainen. Ascent sequences and upper triangular matrices containing non-negative integers. Electron. J. Combin. 17 (2010), no. 1, #R53 (16pp).[3] A. Claesson, M. Dukes and S. Kitaev. A direct encoding of Stoimenow’s matchings as ascentsequences. Australas. J. Combin. 49 (2011), 47–59.[4] R. P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathemat-ics, 62. Cambridge University Press, Cambridge, 1999.

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