A370373 T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.

1, 6, 3, 27, 30, 6, 108, 205, 90, 10, 405, 1188, 870, 210, 15, 1458, 6279, 6888, 2730, 420, 21, 5103, 31306, 48622, 28308, 7070, 756, 28, 17496, 149985, 318726, 256914, 92988, 16002, 1260, 36, 59049, 698256, 1984950, 2136150, 1054305, 260316, 32760, 1980, 45, 196830

Offset

4,2

Links

Table of n, a(n) for n=4..49.

W. Asakly and Noor Kezil, Counting symmetric and non-symmetric peaks in a set partition, arXiv:2401.01687 [math.CO], 2024.

Formula

T(n,k) = binomial(k-1, 2) * Stirling2(n-1, k) + 2 * Sum_{j=3..k} binomial(j, 3) * Sum_{i=3..n-k} j^(i-3) * Stirling2(n-i, k).

Example

The triangle T(n, k) begins:
   4|    1
   5|    6      3
   6|   27     30      6
   7|  108    205     90     10
   8|  405   1188    870    210     15
   9| 1458   6279   6888   2730    420     21
  10| 5103  31306  48622  28308   7070    756   28
.
T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
The canonical form of a set partition of [n] is an n-tuple indicating the block in which each integer occurs. The non-symmetric peaks in the canonical sequential form are listed:
  (1, 2, 3, 1, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 3, 1, 2) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 3, 1, 3) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 1, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 1, 3, 2) -> 1 non-symmetric peak,  (1, 3, 2)

Maple

T := (n, k) -> binomial(k-1, 2) * Stirling2(n-1, k) + 2 * add(binomial(j, 3) * add(j^(i-3) * Stirling2(n-i, k), i=3..n-k), j = 3..k): seq(print(seq(T(n, k), k = 3..n-1)), n = 4..10);

Mathematica

T[n_, k_] := Binomial[k-1, 2] * StirlingS2[n-1, k] + 2 * Sum[Binomial[j, 3] * Sum[j^(i-3) * StirlingS2[n-i, k], {i, 3, n-k}], {j, 3, k}]; Table[T[n, k], {n, 4, 12}, {k, 3, n-1}]

Prog

(PARI) T(n, k) = binomial(k-1, 2) * stirling(n-1, k, 2) + 2 * sum(j=3, k, binomial(j, 3) * sum(i=3, n-k, j^(i-3) * stirling(n-i, k, 2)));

Crossrefs

Cf. A008277 (Stirling2).

Cf. A373288.

Cf. A027471 (1st column), A033487 (subdiagonal).

Sequence in context: A294032 A088697 A039631 * A355867 A287510 A282418

Adjacent sequences: A370370 A370371 A370372 * A370374 A370375 A370376

Keyword

nonn,tabl

Author

Noor Kezil, Jun 07 2024

Status

approved