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Triangle read by rows: T(n,k) is the number of 2-stack sortable n-permutations with k runs.
8

%I #40 Mar 22 2023 17:45:38

%S 1,1,1,1,4,1,1,10,10,1,1,20,49,20,1,1,35,168,168,35,1,1,56,462,900,

%T 462,56,1,1,84,1092,3630,3630,1092,84,1,1,120,2310,12012,20449,12012,

%U 2310,120,1,1,165,4488,34320,91091,91091,34320,4488,165,1,1,220,8151,87516,340340,529984,340340,87516,8151,220,1

%N Triangle read by rows: T(n,k) is the number of 2-stack sortable n-permutations with k runs.

%C Number of beta(1,0)-trees on n+1 nodes with k leaves.

%C Row sums are given by A000139. - _F. Chapoton_, Nov 17 2015

%C T(n,k) is the number of rooted non-separable planar maps with n+1 edges, k+1 faces and n+2-k vertices. - _Andrew Howroyd_, Mar 29 2021

%H Andrew Howroyd, <a href="/A082680/b082680.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%H M. Bona, <a href="https://arxiv.org/abs/math/9705220">2-stack sortable permutations with a given number of runs</a>, arXiv:math/9705220 [math.CO], 1997.

%H Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, <a href="https://arxiv.org/abs/2303.10986">Refined product formulas for Tamari intervals</a>, arXiv:2303.10986 [math.CO], 2023.

%H Enrica Duchi, Veronica Guerrini, Simone Rinaldi, and Gilles Schaeffer, <a href="https://arxiv.org/abs/1611.04625">Fighting Fish: enumerative properties</a>, arXiv:1611.04625 [math.CO], 2016.

%F T(n, k) = (n+k-1)!*(2*n-k)!/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!).

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 10, 10, 1;

%e 1, 20, 49, 20, 1;

%e 1, 35, 168, 168, 35, 1;

%e 1, 56, 462, 900, 462, 56, 1;

%e 1, 84, 1092, 3630, 3630, 1092, 84, 1;

%e ...

%t Table[(n+k-1)!(2n-k)!/k!/(n+1-k)!/(2k-1)!/(2n-2k+1)!,{n,10},{k,n}]//Flatten (* _Harvey P. Dale_, Jun 10 2020 *)

%o (PARI) T(n, k) = (n+k-1)!*(2*n-k)!/k!/(n+1-k)!/(2*k-1)!/(2*n-2*k+1)! \\ _Andrew Howroyd_, Mar 29 2021

%Y Cf. A000292 (2nd column), A051947 (3rd column).

%Y Cf. A000139 (row sums).

%Y Similar to A008292 and A001263.

%K nonn,tabl,easy

%O 1,5

%A _Ralf Stephan_, May 19 2003

%E Terms a(52) and beyond from _Andrew Howroyd_, Mar 29 2021