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.

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, 0, 1, 6, 55, 544, 4725, 29252, 90921, 0, 1, 7, 78, 995, 12196, 123903, 873964, 3081513, 0, 1, 8, 105, 1644, 26215, 377904, 4368729, 34555880, 136407699, 0

Offset: 0

Author

Geoffrey Critzer, Feb 01 2021

Keywords

• nonn
• tabl

Comments

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.

Examples

			Square array T(n,k) begins:
  1,    1,     1,      1,      1,      1, ...
  0,    1,     2,      3,      4,      5, ...
  0,    3,    10,     21,     36,     55, ...
  0,   19,    92,    255,    544,    995, ...
  0,  211,  1354,   4725,  12196,  26215, ...
  0, 3651, 29252, 123903, 377904, 939155, ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Vol. I, Second Edition, Section 3.13.

Links

• Alois P. Heinz, Antidiagonals n = 0..100, flattened

Crossrefs

Columns k=0-4 give: A000007, A000275, A336271, A336638, A336639.

Rows n=0-2 give: A000012, A001477, A014105.

Main diagonal gives A336665.

Cf. A212855, A192721, A334394.

Programs

Maple

T:= (n, k)-> n!^2*coeff(series(1/BesselJ(0, 2*sqrt(x))^k, x, n+1), x, n):
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 02 2021

Mathematica

nn = 6; B[n_] := n!^2; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Grid

Formula

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.

T(n,k) = (n!)^2 * [x^n] 1/BesselJ(0,2*sqrt(x))^k. - Alois P. Heinz, Feb 02 2021

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