|
%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
|
