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A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.

%I A340986 #32 Jul 11 2025 04:25:39
%S A340986 1,1,0,1,1,0,1,2,3,0,1,3,10,19,0,1,4,21,92,211,0,1,5,36,255,1354,3651,
%T A340986 0,1,6,55,544,4725,29252,90921,0,1,7,78,995,12196,123903,873964,
%U A340986 3081513,0,1,8,105,1644,26215,377904,4368729,34555880,136407699,0
%N A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.
%C A340986 A column rise (cf. A000275) means a pair of adjacent columns within a cell where each entry in the first column is less than the adjacent entry in the second column. The order of the columns cannot change. The cells are allowed to be empty.
%D A340986 R. P. Stanley, Enumerative Combinatorics, Vol. I, Second Edition, Section 3.13.
%H A340986 Alois P. Heinz, <a href="/A340986/b340986.txt">Antidiagonals n = 0..100, flattened</a>
%F A340986 Let E(x) = Sum_{n>=0} x^n/n!^2. Then Sum_{n>=0} T(n,k)*x^n/n!^2 = 1/E(-x)^k.
%F A340986 T(n,k) = (n!)^2 * [x^n] 1/BesselJ(0,2*sqrt(x))^k. - _Alois P. Heinz_, Feb 02 2021
%F A340986 For fixed k>=1, T(n,k) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - _Vaclav Kotesovec_, Jul 11 2025
%e A340986 Square array T(n,k) begins:
%e A340986   1,    1,     1,      1,      1,      1, ...
%e A340986   0,    1,     2,      3,      4,      5, ...
%e A340986   0,    3,    10,     21,     36,     55, ...
%e A340986   0,   19,    92,    255,    544,    995, ...
%e A340986   0,  211,  1354,   4725,  12196,  26215, ...
%e A340986   0, 3651, 29252, 123903, 377904, 939155, ...
%p A340986 T:= (n, k)-> n!^2*coeff(series(1/BesselJ(0, 2*sqrt(x))^k, x, n+1), x, n):
%p A340986 seq(seq(T(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Feb 02 2021
%t A340986 nn = 6; B[n_] := n!^2; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
%t A340986 Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Grid
%Y A340986 Columns k=0-4 give: A000007, A000275, A336271, A336638, A336639.
%Y A340986 Rows n=0-2 give: A000012, A001477, A014105.
%Y A340986 Main diagonal gives A336665.
%Y A340986 Cf. A212855, A192721, A334394.
%K A340986 nonn,tabl
%O A340986 0,8
%A A340986 _Geoffrey Critzer_, Feb 01 2021