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Stack sortable permutations

D. Rotem
1981 Discrete Mathematics  
The class SS, of stack sortable permutations is known to be in l-l correspondence with the set of n-noded binary trees.  ...  We use this corresponden(-e to show that many properties of a binary tree are related to different types of monotonic subsequences in the corresponding permutation.  ...  We denote by SS, the class of permutations of order n which are sortable with a stack. and by SR, the class of permutations of the same order which are reraizabie with a stack.  ... 
doi:10.1016/0012-365x(81)90165-5 fatcat:4537o6knh5antizizaanulltl4

Counting 3-Stack-Sortable Permutations [article]

Colin Defant
2020 arXiv   pre-print
The same method yields a recurrence relation for W_3(n), the number of 3-stack-sortable permutations in S_n.  ...  As a first application, we give a new proof of Zeilberger's formula for the number of 2-stack-sortable permutations in S_n.  ...  In this section, we count 3-stack-sortable permutations by viewing them as preimages of 2-stack-sortable permutations.  ... 
arXiv:1903.09138v3 fatcat:kjfanablv5byhlk7ltqxkf7c7e

Stack words and a bound for 3-stack sortable permutations [article]

Miklos Bona
2020 arXiv   pre-print
We use stack words to find a new, simple proof for the best known upper bound for the number of 3-stack sortable permutations of a given length.  ...  This is the first time that stack words are used to obtain such a result.  ...  Enumerating t-stack sortable permutations for t > 1 is extremely difficult. Let W t (n) be the number of t-stack sortable permutations of length n.  ... 
arXiv:1903.04113v4 fatcat:dtdbwteeyjgydlrchrrayho7ia

Describing West-3-stack-sortable permutations with permutation patterns [article]

Henning Úlfarsson
2012 arXiv   pre-print
We use this method to describe West-3-stack-sortable permutations, that is, permutations that are sorted by three passes through a stack.  ...  We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack.  ...  The author would like to thank Anders Claesson for suggesting the problem of describing West-3-stack-sortable permutations, as well as for help with testing Theorem 4.6 with a computer.  ... 
arXiv:1110.1219v3 fatcat:mgr2gwkstvblnd6kttc3dhdfau

Asymptotics of 3-stack-sortable permutations [article]

Colin Defant, Andrew Elvey Price, Anthony J Guttmann
2020 arXiv   pre-print
We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations.  ...  Let W k (n) be the set of k-stack-sortable permutations in S n .  ...  Introduction A permutation is said to be stack-sortable if, when it is passed through a stack, the result is the increasing permutation. We describe the stack-sorting map below.  ... 
arXiv:2009.10439v3 fatcat:7ugl6tjmuzcuvdirwkdoudgomm

2-stack pushall sortable permutations [article]

Adeline Pierrot
2013 arXiv   pre-print
In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related.  ...  Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and describe all its sortings.  ...  Hence π (i) is 2-stack pushall sortable. 2-stack sortable permutations form a permutation class, so that π (k) must be 2-stack sortable.  ... 
arXiv:1303.4376v1 fatcat:rwyz4vlud5f73ifmsbh3vwpyyu

Generalizations of two-stack-sortable permutations [article]

Dapeng Xu
2002 arXiv   pre-print
In this thesis, we apply the stack sorting operator to r-permutations and construct the functional equation for the generating function of two-stack-sortable k-tuple r-permutations counted by descents  ...  We solve the functional equation and give explicit formulas for the number of two-stack-sortable r-permutations.  ...  TWO-STACK-SORTABLE r-PERMUTATIONS . TWO-STACK-SORTABLE r-PERMUTATIONS . TWO-STACK-SORTABLE r-PERMUTATIONS . TWO-STACK-SORTABLE r-PERMUTATIONS . PARALLEL RESULTS .  ... 
arXiv:math/0209313v2 fatcat:vfpe74rvgrb5pd56pv753qkdy4

Fighting fish and two-stack sortable permutations [article]

Wenjie Fang
2019 arXiv   pre-print
In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations.  ...  sortable permutations and non-separable planar maps.  ...  Given a two-stack sortable permutation π, let S(π) be the result of sorting π once through a stack.  ... 
arXiv:1711.05713v2 fatcat:mwvmog7prbcjblshvxa2rrmyoy

Permutations sortable by two stacks in series [article]

Andrew Elvey-Price, Anthony J Guttmann
2016 arXiv   pre-print
We address the problem of the number of permutations that can be sorted by two stacks in series.  ...  We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further coefficients approximately.  ...  Unfortunately, both these cases correspond to an exponentially small subset of the full set of stack-sortable permutations.  ... 
arXiv:1510.08663v2 fatcat:xvh4gk36szhohi2ubhphzzxmcy

Asymptotics of 3-Stack-Sortable Permutations

Colin Defant, Andrew Elvey Price, Anthony Guttmann
2021 Electronic Journal of Combinatorics  
We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations.  ...  Let W k (n) be the set of k-stack-sortable permutations in S n .  ...  A permutation is called k-stack-sortable if iteratively applying the stack-sorting map to it k times results in the increasing permutation.  ... 
doi:10.37236/10134 fatcat:hi2merl62bgb7fzcxsfoj6juce

k-pop stack sortable permutations and 2-avoidance [article]

Murray Elder, Yoong Kuan Goh
2019 arXiv   pre-print
We consider permutations sortable by k passes through a deterministic pop stack.  ...  Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns.  ...  Recall that by Thm. 3.4] and Marcus-Tardos [8] the function counting the number of permutations of length n in any Av(F ) for F non-empty is either polynomial or exponential.  ... 
arXiv:1911.03104v2 fatcat:hxryzkvdx5ft7eaytiyphm35j4

Permutations sortable by two stacks in parallel and quarter plane walks [article]

Michael Albert, Mireille Bousquet-Mélou
2014 arXiv   pre-print
In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns.  ...  At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns.  ...  Denton [11] has presented an algorithm for this problem whose complexity is O(n 5 2 n ) (for enumerating the sortable permutations of length n).  ... 
arXiv:1312.4487v3 fatcat:jdrroduskbh67hpbrid5fpd6jm

Permutations sortable by n-4 passes through a stack [article]

Anders Claesson, Mark Dukes, Einar Steingrimsson
2009 arXiv   pre-print
We characterise and enumerate permutations that are sortable by n-4 passes through a stack.  ...  We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a polynomial expression.  ...  Let the (stack sorting) complexity of π, denoted ssc π, be the smallest k such that π is k-stack sortable.  ... 
arXiv:0812.0143v2 fatcat:bb56laecqra67oc2bxvv42lk7i

Catalan and Schröder permutations sortable by two restricted stacks [article]

J.-L. Baril, G. Cerbai, C. Khalil, V. Vajnovszki
2020 arXiv   pre-print
In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the (binomial transform  ...  They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns.  ...  A permutation π is sortable using a classical stack (that is, a 21-avoiding stack) if and only if π avoids the pattern 231. Let T be a set of patterns.  ... 
arXiv:2004.01812v2 fatcat:5dphlxwvxvbc7hr7pjbewbxwoa

The generating function of two-stack sortable permutations by descents is real-rooted [article]

Petter Brändén
2012 arXiv   pre-print
The paper is withdrawn since the results are included in arXiv:math/0403364.
arXiv:math/0303149v3 fatcat:3ly4mr3ccjgxlc5ozq6ig7hthi
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