A356266 Partition triangle read by rows, counting reducible permutations with weakly decreasing Lehmer code, refining triangle A356115.

1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 3, 3, 1, 0, 1, 4, 4, 2, 6, 12, 2, 4, 6, 1, 0, 1, 5, 5, 5, 10, 20, 10, 10, 10, 30, 10, 5, 10, 1, 0, 1, 6, 6, 6, 3, 15, 30, 30, 15, 15, 20, 60, 30, 60, 5, 15, 60, 30, 6, 15, 1

Offset: 0

Author

Peter Luschny, Aug 16 2022

Keywords

• nonn
• tabf

Examples

			[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 1, 1;
[4] 0, [1, 2], 1, 1;
[5] 0, [1, 3], [3, 3], 3,  1;
[6] 0, [1, 4, 4], [2,  6, 12], [2,  4],  6,  1;
[7] 0, [1, 5, 5], [5, 10, 20, 10], [10, 10, 30], [10,  5], 10,  1;
[8] 0, [1, 6, 6, 6],[3,15, 30, 30, 15],[15, 20, 60, 30, 60],[5,15,60],[30,6],15,1;
Summing the bracketed terms reduces the triangle to A356115.
		

Links

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

Crossrefs

Cf. A356264, A356115 (reduced), A120588 (row sums).

Programs

SageMath

# uses functions perm_red_stats and reducible from A356264.
@cache
def A356266_row(n: int) -> list[int]:
    if n < 2: return [1]
    return [0] + [v[1] for v in perm_red_stats(n, reducible, weakly_decreasing)]
def A356266(n: int, k: int) -> int:
    return A356266_row(n)[k]
for n in range(8):
    print(A356266_row(n))