A249620 Triangle read by rows: T(m,n) = number of partitions of the multiset with m elements and signature corresponding to n-th integer partition (A194602).

1, 1, 2, 2, 5, 4, 3, 15, 11, 7, 9, 5, 52, 36, 21, 26, 12, 16, 7, 203, 135, 74, 92, 38, 52, 19, 66, 29, 31, 11, 877, 566, 296, 371, 141, 198, 64, 249, 98, 109, 30, 137, 47, 57, 15, 4140, 2610, 1315, 1663, 592, 850, 250, 1075, 392, 444, 105, 560

Offset: 0

Author

Tilman Piesk, Nov 04 2014

Keywords

• nonn
• tabf

Comments

This triangle shows the same numbers in each row as A129306 and A096443, but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.

Row lengths: A000041 (partition numbers), Row sums: A035310

Columns: 0: A000110 (Bell), 1: A035098 (near-Bell), 2: A169587, 4: A169588

Last in row: end-1: A091437, end: A000041 (partition numbers)

The rightmost columns form a reflected version of the triangle A126442:

n 0 1 2 4 6 10 14 21 (A000041(1,2,3...)-1)

m

1 1

2 2 2

3 5 4 3

4 15 11 7 5

5 52 36 21 12 7

6 203 135 74 38 19 11

7 877 566 296 141 64 30 15

8 4140 2610 1315 592 250 105 45 22

A249619 shows the number of permutations of the same multisets.

Examples

			See "The T(5,2)=21 partitions of {1,1,1,2,3}" link. Similar links for m=1..8 are in "Partitions of multisets" (Wikiversity).
Triangle begins:
  n     0    1   2   3   4   5   6   7   8   9  10
m
0       1
1       1
2       2    2
3       5    4   3
4      15   11   7   9   5
5      52   36  21  26  12  16   7
6     203  135  74  92  38  52  19  66  29  31  11
		

Links

• Tilman Piesk, Triangle rows m=0..8, flattened.
• Tilman Piesk, Partitions of multisets (Wikiversity)
• Tilman Piesk, The T(5,2)=21 partitions of {1,1,1,2,3}
• Tilman Piesk, PHP code used to calculate the examples

Crossrefs

Cf. A129306, A096443, A035310, A194602, A249619, A000041, A126442