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Number of permutations of length n which avoid the patterns 231, 12534.
1

%I #43 Aug 06 2024 17:01:09

%S 1,2,5,14,41,121,355,1032,2973,8496,24111,68017,190885,533294,1484021,

%T 4115186,11375765,31358377,86223943,236540916,647556621,1769374932,

%U 4826148315,13142564449,35736448201,97037995226,263156279525,712795854422,1928547574913

%N Number of permutations of length n which avoid the patterns 231, 12534.

%H Colin Barker, <a href="/A116845/b116845.txt">Table of n, a(n) for n = 1..1000</a>

%H Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, <a href="http://jl.baril.u-bourgogne.fr/bmrs.pdf">Catalan words avoiding a pattern of length four</a>, Univ. de Bourgogne (France, 2024). See p. 3.

%H Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]

%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,17,-7,1).

%F G.f.: x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2). [restored by _Michael D. Weiner_, Jul 05 2018]

%F a(n) = 7*a(n-1) - 17*a(n-2) + 17*a(n-3) - 7*a(n-4) + a(n-5) for n>5. - _Colin Barker_, Oct 19 2017

%F a(n) = 1 + Fibonacci(2*n)/5 + Lucas(2*n - 3)*n/5. - _Vaclav Kotesovec_, Aug 04 2018

%t Rest[CoefficientList[Series[x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2), {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Aug 04 2018 *)

%t RecurrenceTable[{a[1] == 1, a[2] == 2, a[3] == 5, a[4] == 14, a[5] == 41, a[n] == 7*a[n-1] - 17*a[n-2] + 17*a[n-3] - 7*a[n-4] + a[n-5]}, a, {n, 1, 30}] (* _Vaclav Kotesovec_, Aug 04 2018 *)

%t Table[1 + Fibonacci[2*n]/5 + LucasL[2*n - 3]*n/5, {n, 1, 30}] (* _Vaclav Kotesovec_, Aug 04 2018 *)

%o (PARI) Vec(x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2) + O(x^40)) \\ _Colin Barker_, Oct 19 2017

%Y Cf. A059502 (first differences).

%K nonn,easy

%O 1,2

%A _Lara Pudwell_, Feb 26 2006