login
A356261
Partition triangle read by rows, counting irreducible permutations with weakly decreasing Lehmer code, refining triangle A119308.
1
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
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Aug 16 2022
STATUS
approved