A357079 Triangle read by rows. T(n, k) = A356265(n, k) + A357078(n, k) for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 9, 12, 2, 1, 0, 49, 37, 31, 2, 1, 0, 329, 149, 176, 63, 2, 1, 0, 2561, 794, 853, 702, 127, 2, 1, 0, 22369, 5599, 3836, 5709, 2549, 255, 2, 1, 0, 216225, 47275, 18422, 37609, 33949, 8886, 511, 2, 1, 2291457, 451176, 107535, 218506, 344670, 184653, 29777, 1023, 2, 1

Offset: 0

Author

Peter Luschny, Sep 11 2022

Keywords

• nonn
• tabl

Comments

The triangle is the termwise sum of a refinement of the number of irreducible permutations A357078, and A356265, which is a refinement of the number of reducible permutations.

Examples

			Triangle T(n, k) starts:                                 [Row sums]
[0] 1;                                                       [1]
[1] 0,     1;                                                [1]
[2] 0,     1,      1;                                        [2]
[3] 0,     3,      2,     1;                                 [6]
[4] 0,     9,     12,     2,     1;                          [24]
[5] 0,    49,     37,    31,     2,     1;                   [120]
[6] 0,   329,    149,   176,    63,     2,    1;             [720]
[7] 0,   2561,   794,   853,   702,   127,    2,   1;        [5040]
[8] 0,  22369,  5599,  3836,  5709,  2549,  255,   2, 1;     [40320]
[9] 0, 216225, 47275, 18422, 37609, 33949, 8886, 511, 2, 1;  [362880]
		

Crossrefs

Cf. A356265, A357078, A059438, A003319.

Programs

SageMath

def A357079_triangle(dim):
    T = A357078_triangle(dim)
    return [[0^n] + [T[n][k+1] + A356265_row(n)[k]
           for k in range(n)] for n in range(dim)]
A357079_triangle(9)