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

0,6

Links

Table of n, a(n) for n=0..66.

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

Example

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.

Prog

(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))

Crossrefs

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

Sequence in context: A240808 A263142 A025253 * A356262 A281228 A284575

Adjacent sequences: A356258 A356259 A356260 * A356262 A356263 A356264

Keyword

nonn,tabf

Author

Peter Luschny, Aug 16 2022

Status

approved