scholarly journals Refined Restricted Permutations

2002 ◽  
Vol 6 (3) ◽  
pp. 427-444 ◽  
Author(s):  
Aaron Robertson ◽  
Dan Saracino ◽  
Doron Zeilberger
2010 ◽  
Vol 4 (1) ◽  
pp. 119-135 ◽  
Author(s):  
Vladimir Baltic

Let p be a permutation of the set Nn={1, 2,...,n}. We introduce techniques for counting N(n; k, r, I), the number of Lehmer?s strongly restricted permutations of Nn satisfying the conditions ?k ? p(i)?i ? r (for arbitrary natural numbers k and r) and p(i) ? i I (for some set I). We show that N(n; 1, r, ?) is the Fibonacci (r + 1)-step number.


10.37236/440 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christian Houdré ◽  
Ricardo Restrepo

Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a uniform random permutation $\tau\in\left[ n\right] $. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}\left( \tau\right) $. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.


10.37236/1686 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Astrid Reifegerste

We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns $\tau_1\cdots\tau_{m-2}m(m-1)$ and $\tau_1\cdots\tau_{m-2}(m-1)m$ in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns $\tau_1\cdots\tau_{m-2}m(m-1)\in{\cal S}_m$ and those which avoid each of the patterns $\tau_1\cdots\tau_{m-2}(m-1)m\in{\cal S}_m$. For $m=3$ this correspondence coincides with the bijection given by Simion and Schmidt in [Europ. J. Combin. 6 (1985), 383-406].


2012 ◽  
Vol 22 (2) ◽  
pp. 183-198
Author(s):  
Vladimir Baltic

In this paper, we use the finite state automata to count the number of restricted permutations and the number of restricted variations. For each type of restricted permutations, we construct a finite state automaton able to recognize and enumerate them. We, also, discuss how it encompasses the other known methods for enumerating permutations with restricted position, and in one case, we establish connections with some other combinatorial structures, such as subsets and compositions.


2006 ◽  
Vol 154 (11) ◽  
pp. 1593-1605 ◽  
Author(s):  
Toufik Mansour ◽  
Eva Y.P. Deng ◽  
Rosena R.X. Du

1985 ◽  
Vol 6 (4) ◽  
pp. 383-406 ◽  
Author(s):  
Rodica Simion ◽  
Frank W. Schmidt

2015 ◽  
Vol 187 ◽  
pp. 82-90
Author(s):  
Kenneth Edwards ◽  
Michael A. Allen

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović

International audience We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.


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