A356261 Partition triangle read by rows, counting irreducible permutations with weakly decreasing Lehmer code, refining triangle A119308.

1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 5, 1, 0, 2, 2, 7, 7, 9, 1, 0, 2, 2, 1, 9, 18, 3, 16, 24, 14, 1, 0, 2, 2, 2, 11, 22, 11, 11, 25, 75, 25, 30, 60, 20, 1, 0, 2, 2, 2, 1, 13, 26, 26, 13, 13, 36, 108, 54, 108, 9, 55, 220, 110, 50, 125, 27, 1

Offset: 0

Author

Peter Luschny, Aug 16 2022

Keywords

• nonn
• tabf

Examples

			Partition table T(n, k) begins:
[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 2, 1;
[4] 0, [2, 1],  5,  1;
[5] 0, [2, 2], [7,  7],   9,  1;
[6] 0, [2, 2,  1], [9,   18, 3], [16, 24], 14,    1;
[7] 0, [2, 2,  2], [11,  22, 11, 11], [25, 75,  25], [30, 60],  20, 1;
[8] 0, [2, 2, 2, 1],[13, 26, 26, 13, 13],[36, 108, 54, 108,9],[55, 220, 110],[50, 125], 27, 1;
Summing the bracketed terms reduces the triangle to A119308.
		

Links

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

Crossrefs

Cf. A356264, A119308 (reduced), A071724 (row sums).

Programs

SageMath

# using function perm_red_stats and reducible from A356264
def weakly_decreasing(L: list[int]) -> bool:
    return all(x >= y for x, y in zip(L, L[1:]))
@cache
def A356261_row(n: int) -> list[int]:
    if n < 2: return [1]
    return [0] + [v[1] for v in perm_red_stats(n, irreducible, weakly_decreasing)]
def A356261(n: int, k: int) -> int:
    return A356261_row(n)[k]
for n in range(8):
    print([n], A356261_row(n))