A393395 - OEIS

%I #19 Feb 23 2026 09:28:51

%S 1,1,2,6,24,116,628,3636,21956,136428,865700,5583580,36490740,

%T 241105516,1607891780,10808452348,73159943124,498215170508,

%U 3411058716964,23465785541468,162121010290996,1124393414877996,7825541946451972,54637557152397884,382586828736801940,2686126809776641932

%N Number of permutations of length n sortable by a (2,2)-stack.

%C A (2,2)-stack allows pushing into either of the top two positions and popping from either of the top two positions.

%C This class is defined by avoiding the permutations 23451, 23541, 32451, 32541, 245163, 246153, 425163, 426153.

%H Andrew Howroyd, <a href="/A393395/b393395.txt">Table of n, a(n) for n = 0..500</a>

%H M. D. Atkinson, <a href="https://doi.org/10.1017/S096354839800354X">Generalized stack permutations</a>, Combin. Probab. Comput. 7 (1998), no. 3, 239-246.

%F G.f. f(x) satisfies 2*x*f(x)^3 - (2*x+3)*f(x)^2 - (x-7)*f(x) - 4 = 0.

%F G.f.: 1 + Series_Reversion(x*(3*x - 1)/((x + 1)*(2*x^2 + 2*x - 1))). - _Andrew Howroyd_, Feb 18 2026

%F a(n) ~ sqrt(s*(1 + 2*s - 2*s^2) / (Pi*(3 + 2*r - 6*r*s))) / (2*n^(3/2) * r^(n - 1/2)), where r = (52 + sqrt(10531)*cos((2*Pi + arccos(1080647/10531^(3/2)))/3))/3 = 0.1338790671510627252896614474332139209739178... and s = (8 + sqrt(94)*cos((4*Pi + arccos(-191*sqrt(2/47)/47))/3))/9 = 1.2393827937375600536930234428899841975948603... - _Vaclav Kotesovec_, Feb 18 2026

%o (PARI) seq(n)={Vec(1 + serreverse(x*(3*x - 1)/((x + 1)*(2*x^2 + 2*x - 1)) + O(x*x^n)))} \\ _Andrew Howroyd_, Feb 18 2026

%K nonn

%O 0,3

%A _Vincent Vatter_, Feb 17 2026