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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.

%I A373288 #101 Jun 07 2024 07:25:55
%S A373288 1,3,2,8,12,3,20,54,30,4,48,215,205,60,5,112,799,1185,580,105,6,256,
%T A373288 2842,6230,4585,1365,168,7,576,9812,30828,32256,14140,2828,252,8,1280,
%U A373288 33165,146355,210378,128037,37170,5334,360,9,2816,110361,674535,1301860,1060815,420756,86730,9360,495,10
%N 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.
%H A373288 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 A373288 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 A373288 The triangle T(n, k) begins:
%e A373288    3|   1
%e A373288    4|   3      2
%e A373288    5|   8     12      3
%e A373288    6|  20     54     30      4
%e A373288    7|  48    215    205     60      5
%e A373288    8| 112    799   1185    580    105      6
%e A373288    9| 256   2842   6230   4585   1365    168      7
%e A373288   10| 576   9812  30828  32256  14140   2828    252      8
%e A373288 .
%e A373288 T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
%e A373288 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 A373288   (1, 2, 1, 1, 3) -> 1 symmetric peak   (1, 2, 1)
%e A373288   (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
%e A373288   (1, 2, 1, 2, 3) -> 1 symmetric peak,  (1, 2, 1)
%e A373288   (1, 2, 1, 3, 2) -> 1 symmetric peak,  (1, 2, 1)
%e A373288   (1, 2, 1, 3, 3) -> 1 symmetric peak,  (1, 2, 1)
%e A373288   (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
%e A373288   (1, 2, 2, 3, 2) -> 1 symmetric peak,  (2, 3, 2)
%e A373288   (1, 2, 3, 2, 1) -> 1 symmetric peak,  (2, 3, 2)
%e A373288   (1, 2, 3, 2, 2) -> 1 symmetric peak,  (2, 3, 2)
%e A373288   (1, 2, 3, 2, 3) -> 1 symmetric peak,  (2, 3, 2).
%p A373288 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 A373288 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 A373288 Table[T[n, k], {n, 3, 12}, {k, 2, n-1}] // Flatten
%o A373288 (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 A373288 Cf. A008277 (Stirling2).
%Y A373288 Cf. A001792 (1st column), A027480 (subdiagonal).
%K A373288 nonn,tabl
%O A373288 3,2
%A A373288 _W. Asakly_, Jun 01 2024