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

0,6

Links

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

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

Example

[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.

Prog

(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))

Crossrefs

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

Sequence in context: A175528 A163940 A112340 * A037186 A343239 A004483

Adjacent sequences: A356261 A356262 A356263 * A356265 A356266 A356267

Keyword

nonn,tabf

Author

Peter Luschny, Aug 05 2022

Status

approved