A356265 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations (A356264). T(n, k) for n >= 1 and 0 <= k < n.

0, 1, 0, 1, 2, 0, 1, 8, 2, 0, 1, 21, 25, 2, 0, 1, 49, 152, 55, 2, 0, 1, 106, 697, 670, 117, 2, 0, 1, 223, 2756, 5493, 2509, 243, 2, 0, 1, 459, 9966, 36105, 33669, 8838, 497, 2, 0, 1, 936, 34095, 206698, 342710, 184305, 29721, 1007, 2, 0

Offset: 1

Author

Peter Luschny, Aug 16 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:                        [Row sums]
[1] [0]                                            [0]
[2] [1,   0]                                       [1]
[3] [1,   2,    0]                                 [3]
[4] [1,   8,    2,     0]                          [11]
[5] [1,  21,   25,     2,     0]                   [49]
[6] [1,  49,  152,    55,     2,    0]             [259]
[7] [1, 106,  697,   670,   117,    2,   0]        [1593]
[8] [1, 223, 2756,  5493,  2509,  243,   2, 0]     [11227]
[9] [1, 459, 9966, 36105, 33669, 8838, 497, 2, 0]  [89537]
		

Links

• Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Crossrefs

Cf. A356264 (partitions), A356291 (row sums).

Programs

SageMath

# uses function A356264_row
@cache
def Pn(n: int, k: int) -> int:
    if k == 0: return 0
    if n == 0 or k == 1: return 1
    return Pn(n, k - 1) + Pn(n - k, k) if k <= n else Pn(n, k - 1)
def reduce_parts(fun, n: int) -> list[int]:
    funn: list[int] = fun(n)
    return [sum(funn[Pn(n, k):Pn(n, k + 1)]) for k in range(n)]
def reduce_partition_triangle(fun, n: int) -> list[list[int]]:
    return [reduce_parts(fun, k) for k in range(1, n)]
def A356265_row(n: int) -> list[int]:
    return reduce_partition_triangle(A356264_row, n+1)[n-1]
for n in range(1, 8):
    print(A356265_row(n))