OFFSET
0,5
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 22 2020
STATUS
approved