A212433
Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac <--> cba, where a
1, 1, 2, 3, 13, 71, 470, 3497, 29203, 271500, 2786711, 31322803, 382794114, 5054810585, 71735226535, 1088920362030, 17607174571553, 302143065676513, 5484510055766118, 104999034898520903, 2114467256458136473, 44682676397748896010, 988663144904696100347
Offset: 0
Keywords
Examples
From _Alois P. Heinz_, Jun 23 2012: (Start)
a(3) = 3: {123, 132, 213, 321}, {231}, {312}.
a(4) = 13: {1234, 1243, 1324, 1432, 2134, 3214}, {1342}, {1423}, {2143}, {2314}, {2341, 2431, 3241, 4123, 4132, 4213, 4321}, {2413}, {3124}, {3142}, {3412}, {3421}, {4231}, {4312}.
a(5) = 71: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 21345, 21354, 21435, 21543, 32145, 32154}, {12453}, ..., {53412}, {53421}, {54231}.
(End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
- Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
- S. Linton, J. Propp, T. Roby, and J. West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, arXiv:1111.3920, 2011 [math.CO]
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*((1-x^2)^2/(1-x^3)-x^2))^k)) \\ Seiichi Manyama, Feb 25 2024
Formula
G.f.: Sum_{k>=0} k! * ( x * ((1-x^2)^2/(1-x^3) - x^2) )^k. - Seiichi Manyama, Feb 25 2024
Extensions
a(8)-a(9) from Alois P. Heinz, Jun 23 2012
a(10)-a(22) from Alois P. Heinz, Apr 14 2021