A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 8, 1, 0, 5, 85, 50, 10, 1, 0, 6, 300, 280, 75, 12, 1, 0, 7, 1071, 1540, 525, 105, 14, 1, 0, 8, 3976, 8456, 3570, 840, 140, 16, 1, 0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Example

Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 1]
3: [0, 3, 6,     1]
4: [0, 4, 24,    8,     1]
5: [0, 5, 85,    50,    10,    1]
6: [0, 6, 300,   280,   75,    12,   1]
7: [0, 7, 1071,  1540,  525,   105,  14,   1]
8: [0, 8, 3976,  8456,  3570,  840,  140,  16,  1]
9: [0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1]
.
T(4,0) = 0  = 0*card({})
T(4,1) = 4  = 4*card({1|2|3|4}).
T(4,2) = 24 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
            + 2*card({12|34, 13|24, 14|23}).
T(4,3) = 8  = 2*card({123|4, 124|3, 134|2, 1|234}).
T(4,4) = 1  = 1*card({1234}).
.
Seen as the projection of a 2-dimensional statistic this is, for n = 6:
[  0   0    0     0     0    0   0]
[  0   0    0     0     0    0   6]
[  0   0    0    45   180   75   0]
[  0   0   20   180    80    0   0]
[  0   0   30    45     0    0   0]
[  0   0   12     0     0    0   0]
[  0   1    0     0     0    0   0]
The row sum projection gives row 6 of this triangle, and the column sum projection gives [0, 1, 62, 270, 260, 75, 6], which appears in a decapitated version as row 5 in A321331.

Prog

(SageMath)
def A339030Row(n):
    if n == 0: return [1]
    M = matrix(n + 1)
    for k in (1..n):
        for p in SetPartitions(n):
            if p.max_block_size() == k:
                M[k, len(p)] += p.cardinality()
    return [sum(M[k, j] for j in (0..n)) for k in (0..n)]
for n in (0..9): print(A339030Row(n))

Crossrefs

Cf. A005493 with 1 prepended are the row sums.

Cf. A080510, A262071, A321331.

Sequence in context: A105546 A342237 A367267 * A059297 A267222 A077874

Adjacent sequences: A339027 A339028 A339029 * A339031 A339032 A339033

Keyword

nonn,tabl

Author

Peter Luschny, Nov 22 2020

Status

approved