A356264 Partition triangle read by rows, counting reducible permutations, refining triangle A356265.

0, 0, 1, 0, 1, 2, 0, 1, 5, 3, 2, 0, 1, 9, 12, 15, 10, 2, 0, 1, 14, 23, 12, 47, 94, 11, 31, 24, 2, 0, 1, 20, 38, 48, 113, 293, 154, 137, 183, 409, 78, 63, 54, 2, 0, 1, 27, 60, 87, 49, 227, 738, 883, 451, 457, 670, 2157, 1007, 1580, 79, 605, 1520, 384, 127, 116, 2, 0

Offset: 0

Author

Peter Luschny, Aug 05 2022

Keywords

• nonn
• tabf

Examples

			[0] 0;
[1] 0;
[2] 1, 0;
[3] 1, 2, 0;
[4] 1, [5, 3], 2, 0;
[5] 1, [9, 12], [15,  10],   2,  0;
[6] 1, [14, 23, 12], [ 47,  94, 11], [31,   24],   2,  0;
[7] 1, [20, 38, 48], [113, 293, 154, 137], [183, 409, 78], [63, 54], 2, 0;
Summing the bracketed terms reduces the triangle to A356265.
		

Links

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

Crossrefs

Cf. A356265 (reduced), A356262, A356263, A356291 (row sums).

Programs

SageMath

import collections
def reducible(p) -> bool:  # p is a Sage-Permutation
    return any(i for i in range(1, p.size())
                  if all(p(j) < p(k)
                      for j in range(1, i + 1)
                          for k in range(i + 1, p.size() + 1) ) )
def void(L) -> bool: return True
def perm_red_stats(n: int, part_costraint, lehmer_constraint):
    res = collections.defaultdict(int)
    for p in Permutations(n):
        if not part_costraint(p): continue
        l: list[int] = p.to_lehmer_code()
        if lehmer_constraint(l):
            c: list[int] = [l.count(i) for i in range(len(p)) if i in l]
            res[Partition(reversed(sorted(c)))] += 1
    return sorted(res.items(), key=lambda x: len(x[0]))
@cache
def A356264_row(n: int) -> list[int]:
    if n < 2: return [0]
    return [v[1] for v in perm_red_stats(n, reducible, void)] + [0]
def A356264(n: int, k: int) -> int:
    return A356264_row(n)[k]
for n in range(0, 8): print(A356264_row(n))