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.
1, 6, 3, 27, 30, 6, 108, 205, 90, 10, 405, 1188, 870, 210, 15, 1458, 6279, 6888, 2730, 420, 21, 5103, 31306, 48622, 28308, 7070, 756, 28, 17496, 149985, 318726, 256914, 92988, 16002, 1260, 36, 59049, 698256, 1984950, 2136150, 1054305, 260316, 32760, 1980, 45, 196830
Offset: 4
Examples
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)
Links
- W. Asakly and Noor Kezil, Counting symmetric and non-symmetric peaks in a set partition, arXiv:2401.01687 [math.CO], 2024.
Programs
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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);
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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}] -
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)));
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).