%I #18 Oct 30 2021 07:17:54

%S 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,

%T 66,29,31,11,877,566,296,371,141,198,64,249,98,109,30,137,47,57,15,

%U 4140,2610,1315,1663,592,850,250,1075,392,444,105,560

%N 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).

%C 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.

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

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

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

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

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

%C m

%C 1 1

%C 2 2 2

%C 3 5 4 3

%C 4 15 11 7 5

%C 5 52 36 21 12 7

%C 6 203 135 74 38 19 11

%C 7 877 566 296 141 64 30 15

%C 8 4140 2610 1315 592 250 105 45 22

%C A249619 shows the number of permutations of the same multisets.

%H Tilman Piesk, <a href="/A249620/b249620.txt">Triangle rows m=0..8, flattened.</a>

%H Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Partitions_of_multisets">Partitions of multisets</a> (Wikiversity)

%H Tilman Piesk, <a href="http://pastebin.com/HuiVhmrP">The T(5,2)=21 partitions of {1,1,1,2,3}</a>

%H Tilman Piesk, <a href="http://pastebin.com/jHr76sFa">PHP code used to calculate the examples</a>

%e 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).

%e Triangle begins:

%e n 0 1 2 3 4 5 6 7 8 9 10

%e m

%e 0 1

%e 1 1

%e 2 2 2

%e 3 5 4 3

%e 4 15 11 7 9 5

%e 5 52 36 21 26 12 16 7

%e 6 203 135 74 92 38 52 19 66 29 31 11

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

%K nonn,tabf

%O 0,3

%A _Tilman Piesk_, Nov 04 2014