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

0,8

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.

References

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

Links

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

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

Example

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, ...

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

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.

Sequence in context: A294046 A320079 A349971 * A340798 A355427 A122078

Adjacent sequences: A340983 A340984 A340985 * A340987 A340988 A340989

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 01 2021

Status

approved