OFFSET
4,2
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Noor Kezil, Jun 07 2024
STATUS
approved