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A319251 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k descents such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.

%I A319251 #12 Nov 17 2020 14:25:21
%S A319251 1,1,1,1,4,1,1,8,8,1,1,13,28,13,1,1,19,70,70,19,1,1,26,145,250,145,26,
%T A319251 1,1,34,266,700,700,266,34,1,1,43,448,1666,2548,1666,448,43,1,1,53,
%U A319251 708,3528,7644,7644,3528,708,53,1
%N A319251 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k descents such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.
%C A319251 Rows are symmetric, so T(n,k) = T(n,n-1-k).
%C A319251 T(n,k) is the number of permutations of [n] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421 and have k descents.
%H A319251 Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%F A319251 T(n,k) = N(n,k+1) + Sum_{i=1..n-2} Sum_{j=1..m} N(n-i-1,j) * N(i,k-j+1), where N(i,j) = (1/i) * binomial(i,j) * binomial(i,j-1) are the Narayana numbers given in A001263.
%F A319251 From _Vladimir Kruchinin_, Nov 16 2020: (Start)
%F A319251 G.f.: A001263(x,y) + x*A001263(x,y)^2.
%F A319251 T(n,k) = N(n,k) +2*C(n-1,k-2)*C(n-1,k)/(n-1). (End)
%e A319251 Triangle begins:
%e A319251   1,
%e A319251   1, 1,
%e A319251   1, 4, 1,
%e A319251   1, 8, 8, 1,
%e A319251   1, 13, 28, 13, 1,
%e A319251   1, 19, 70, 70, 19, 1
%e A319251   ...
%t A319251 Flatten[Table[Table[(1/n) Binomial[n, m + 1] Binomial[n, m] + Sum[Sum[(1/(n - i - 1)) Binomial[n - i - 1, j] Binomial[n - i - 1, j - 1] (1/i) Binomial[i, m - j + 1] Binomial[i, m - j], {j, 1, m}], {i, 1, n - 2}], {m, 0, n - 1}], {n, 1, 10}]]
%Y A319251 Row sums give A071721. Cf. A001263, A319252.
%K A319251 easy,nonn,tabl
%O A319251 1,5
%A A319251 _Colin Defant_, Sep 15 2018