%I #101 Jun 07 2024 07:25:55

%S 1,3,2,8,12,3,20,54,30,4,48,215,205,60,5,112,799,1185,580,105,6,256,

%T 2842,6230,4585,1365,168,7,576,9812,30828,32256,14140,2828,252,8,1280,

%U 33165,146355,210378,128037,37170,5334,360,9,2816,110361,674535,1301860,1060815,420756,86730,9360,495,10

%N 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.

%H W. Asakly and Noor Kezil, <a href="https://arxiv.org/abs/2401.01687">Counting symmetric and non-symmetric peaks in a set partition</a>, arXiv:2401.01687 [math.CO], 2024.

%F 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).

%e The triangle T(n, k) begins:

%e 3| 1

%e 4| 3 2

%e 5| 8 12 3

%e 6| 20 54 30 4

%e 7| 48 215 205 60 5

%e 8| 112 799 1185 580 105 6

%e 9| 256 2842 6230 4585 1365 168 7

%e 10| 576 9812 30828 32256 14140 2828 252 8

%e .

%e T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:

%e 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:

%e (1, 2, 1, 1, 3) -> 1 symmetric peak (1, 2, 1)

%e (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)

%e (1, 2, 1, 2, 3) -> 1 symmetric peak, (1, 2, 1)

%e (1, 2, 1, 3, 2) -> 1 symmetric peak, (1, 2, 1)

%e (1, 2, 1, 3, 3) -> 1 symmetric peak, (1, 2, 1)

%e (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)

%e (1, 2, 2, 3, 2) -> 1 symmetric peak, (2, 3, 2)

%e (1, 2, 3, 2, 1) -> 1 symmetric peak, (2, 3, 2)

%e (1, 2, 3, 2, 2) -> 1 symmetric peak, (2, 3, 2)

%e (1, 2, 3, 2, 3) -> 1 symmetric peak, (2, 3, 2).

%p 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

%t 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}];

%t Table[T[n, k], {n, 3, 12}, {k, 2, n-1}] // Flatten

%o (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

%Y Cf. A008277 (Stirling2).

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

%K nonn,tabl

%O 3,2

%A _W. Asakly_, Jun 01 2024