OFFSET
3,2
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) = (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
KEYWORD
nonn,tabl
AUTHOR
W. Asakly, Jun 01 2024
STATUS
approved