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A370373 T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.

%I A370373 #45 Jun 13 2024 14:57:52
%S A370373 1,6,3,27,30,6,108,205,90,10,405,1188,870,210,15,1458,6279,6888,2730,
%T A370373 420,21,5103,31306,48622,28308,7070,756,28,17496,149985,318726,256914,
%U A370373 92988,16002,1260,36,59049,698256,1984950,2136150,1054305,260316,32760,1980,45,196830
%N A370373 T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.
%H A370373 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 A370373 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).
%e A370373 The triangle T(n, k) begins:
%e A370373    4|    1
%e A370373    5|    6      3
%e A370373    6|   27     30      6
%e A370373    7|  108    205     90     10
%e A370373    8|  405   1188    870    210     15
%e A370373    9| 1458   6279   6888   2730    420     21
%e A370373   10| 5103  31306  48622  28308   7070    756   28
%e A370373 .
%e A370373 T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
%e A370373 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:
%e A370373   (1, 2, 3, 1, 1) -> 1 non-symmetric peak,  (2, 3, 1)
%e A370373   (1, 2, 3, 1, 2) -> 1 non-symmetric peak,  (2, 3, 1)
%e A370373   (1, 2, 3, 1, 3) -> 1 non-symmetric peak,  (2, 3, 1)
%e A370373   (1, 2, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
%e A370373   (1, 1, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
%e A370373   (1, 2, 1, 3, 2) -> 1 non-symmetric peak,  (1, 3, 2)
%p A370373 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);
%t A370373 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}]
%o A370373 (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)));
%Y A370373 Cf. A008277 (Stirling2).
%Y A370373 Cf. A373288.
%Y A370373 Cf. A027471 (1st column), A033487 (subdiagonal).
%K A370373 nonn,tabl
%O A370373 4,2
%A A370373 _Noor Kezil_, Jun 07 2024