cp's OEIS Frontend

A319030 Triangle read by rows: T(n,k) is the number of permutations pi of [n] such that pi has k+1 valleys and s(pi) avoids the patterns 132 and 321, where s is West's stack-sorting map (0 <= k <= floor((n-1)/2)).

%I A319030 #16 Sep 14 2018 12:32:17
%S A319030 1,2,4,2,8,14,16,64,8,32,240,92,64,800,624,34,128,2464,3248,534,256,
%T A319030 7168,14336,4736,144,512,19968,56448,31200,2852,1024,53760,204288,
%U A319030 169920,31120,604,2048,140800,692736,808896,247280,14412
%N A319030 Triangle read by rows: T(n,k) is the number of permutations pi of [n] such that pi has k+1 valleys and s(pi) avoids the patterns 132 and 321, where s is West's stack-sorting map (0 <= k <= floor((n-1)/2)).
%C A319030 Row sums give A319028.
%H A319030 Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%F A319030 G.f.: G(x,y) + x^3*y*((d/dx)G(x,y))^2, where G(x,y) = (1 - 2x - sqrt((1-2x)^2 - 4x^2*y))/(2x*y) is the generating function of A091894.
%e A319030 Triangle begins:
%e A319030     1,
%e A319030     2,
%e A319030     4,    2,
%e A319030     8,   14,
%e A319030    16,   64,    8,
%e A319030    32,  240,   92,
%e A319030    64,  800,  624,  34,
%e A319030   128, 2464, 3248, 534,
%e A319030   ...
%t A319030 DeleteCases[Flatten[CoefficientList[Series[(1 - 2 x - Sqrt[(1 - 2 x)^2 - 4 x^2 y])/(2 x*y) + x^3*y (D[(1 - 2 x - Sqrt[(1 - 2 x)^2 - 4 x^2 y])/(2 x*y), x])^2, {x, 0, 10}], {x, y}]], 0]
%Y A319030 Cf. A319028, A091894.
%K A319030 easy,nonn,tabf
%O A319030 1,2
%A A319030 _Colin Defant_, Sep 10 2018