A373288 T(n, k) is the total number of symmetric peaks in all partitions of n with exactly k blocks, n >= 3, 2 <= k <= n-1.

1, 3, 2, 8, 12, 3, 20, 54, 30, 4, 48, 215, 205, 60, 5, 112, 799, 1185, 580, 105, 6, 256, 2842, 6230, 4585, 1365, 168, 7, 576, 9812, 30828, 32256, 14140, 2828, 252, 8, 1280, 33165, 146355, 210378, 128037, 37170, 5334, 360, 9, 2816, 110361, 674535, 1301860, 1060815, 420756, 86730, 9360, 495, 10

Offset

3,2

Links

Table of n, a(n) for n=3..57.

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

Example

The triangle T(n, k) begins:
   3|   1
   4|   3      2
   5|   8     12      3
   6|  20     54     30      4
   7|  48    215    205     60      5
   8| 112    799   1185    580    105      6
   9| 256   2842   6230   4585   1365    168      7
  10| 576   9812  30828  32256  14140   2828    252      8
.
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 symmetric peaks in the canonical sequential form are listed:
  (1, 2, 1, 1, 3) -> 1 symmetric peak   (1, 2, 1)
  (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
  (1, 2, 1, 2, 3) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 2) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 3) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
  (1, 2, 2, 3, 2) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 1) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 2) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 3) -> 1 symmetric peak,  (2, 3, 2).

Maple

T := (n, k) -> (k-1) * Stirling2(n-1, k) + add(binomial(j, 2) * add(j^(i-3) * Stirling2(n-i, k), i=3..n-k), j = 2..k): seq(print(seq(T(n, k), k = 2..n-1)), n = 3..10);  # Peter Luschny, Jun 06 2024

Mathematica

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

Prog

(PARI) T(n, k) = (k-1) * stirling(n-1, k, 2) + sum(j=2, k, binomial(j, 2) * sum(i=3, n-k, j^(i-3) * stirling(n-i, k, 2))); \\ Michel Marcus, Jun 06 2024

Crossrefs

Cf. A008277 (Stirling2).

Cf. A001792 (1st column), A027480 (subdiagonal).

Sequence in context: A095013 A094188 A088551 * A301903 A165660 A171634

Adjacent sequences: A373285 A373286 A373287 * A373289 A373290 A373291

Keyword

nonn,tabl

Author

W. Asakly, Jun 01 2024

Status

approved