Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Hackl, Benjamin; Heuberger, Clemens; Prodinger, Helmut; Wagner, Stephan

doi:10.1007/s00026-016-0330-0

Cited by 6 publications

(5 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In previous papers on BBS's ( [11], [2], [6]), analytic techniques were used to obtain asymptotics, but our techniques reveal a geometric interpretation for the Θ(2 n /n) growth rate.…”

Section: Discussionmentioning

confidence: 94%

See 1 more Smart Citation

The bidirectional ballot polytope

Miller,

Peterson,

Sprunger

et al. 2017

Preprint

View full text Add to dashboard Cite

A bidirectional ballot sequence (BBS) is a finite binary sequence with the property that every prefix and suffix contains strictly more ones than zeros. BBS's were introduced by Zhao, and independently by Bosquet-Mélou and Ponty as (1, 1)-culminating paths. Both sets of authors noted the difficulty in counting these objects, and to date research on bidirectional ballot sequences has been concerned with asymptotics. We introduce a continuous analogue of bidirectional ballot sequences which we call bidirectional gerrymanders, and show that the set of bidirectional gerrymanders form a convex polytope sitting inside the unit cube, which we refer to as the bidirectional ballot polytope. We prove that every (2n − 1)-dimensional unit cube can be partitioned into 2n − 1 isometric copies of the (2n − 1)-dimensional bidirectional ballot polytope. Furthermore, we show that the vertices of this polytope are all also vertices of the cube, and that the vertices are in bijection with BBS's. An immediate corollary is a geometric explanation of the result of Zhao and of Bosquet-Mélou and Ponty that the number of BBS's of length n is Θ(2 n /n).

show abstract

“…In previous papers on BBS's ( [11], [2], [6]), analytic techniques were used to obtain asymptotics, but our techniques reveal a geometric interpretation for the Θ(2 n /n) growth rate.…”

Section: Discussionmentioning

confidence: 94%

“…Additionally in [11], the author conjectured an even finer asymptotic expression for B n . This conjecture was later proved by Hackl, Heuberger, Prodinger and Wagner [6], who refined the asymptotic expression even further using techniques from analytic combinatorics.…”

Section: Introductionmentioning

confidence: 92%

The bidirectional ballot polytope

Miller,

Peterson,

Sprunger

et al. 2017

Preprint

View full text Add to dashboard Cite

show abstract

“…The lattice paths occurring in Corollary 16 have been studied in numerous papers, such as [4,6,13]. The main term of their asymptotic enumeration can be found in [4], while the most precise enumerative result is proved in [6], where these lattice paths are called bidirectional ballot sequences, and the number of such paths of length n is denoted by B n . The result is that…”

Section: The Pattern 132mentioning

confidence: 99%

Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence

Bóna

DeJonge

2020

Electron. J. Combin.

View full text Add to dashboard Cite

We investigate permutations and involutions that avoid a pattern of length three and have a unique longest increasing subsequence (ULIS). We prove an explicit formula for 231-avoiders, we show that the growth rate for 321-avoiding permutations with a ULIS is 4, and prove that their generating function is not rational. We relate the case of 132-avoiders to the existing literature, raising some interesting questions. For involutions, we construct a bijection between 132-avoiding involutions with a ULIS and bidirectional ballot sequences.

show abstract

“…1. For q = 1, Lehmer's determinant plays a role when enumerating lattice paths (Dyck paths) of bounded height, or planar trees of bounded height, see [5,9,7].…”

Section: Remarksmentioning

confidence: 99%