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Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).
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%I #15 Nov 01 2021 03:15:28

%S 1,1,2,1,6,3,1,24,12,4,6,1,120,60,20,30,5,10,1,720,360,120,180,30,60,

%T 6,90,15,20,1,5040,2520,840,1260,210,420,42,630,105,140,7,210,21,35,1,

%U 40320,20160,6720,10080,1680,3360,336,5040,840,1120,56

%N Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

%C This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), 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: A005651

%C Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388

%C Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012

%C The rightmost columns form the triangle A173333:

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

%C m

%C 1 1

%C 2 2 1

%C 3 6 3 1

%C 4 24 12 4 1

%C 5 120 60 20 5 1

%C 6 720 360 120 30 6 1

%C 7 5040 2520 840 210 42 7 1

%C 8 40320 20160 6720 1680 336 56 8 1

%C A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

%H Tilman Piesk, <a href="/A249619/b249619.txt">Triangle rows m=0..25, flattened</a>

%H Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Multisets_with_m_elements_corresponding_to_integer_partitions_n.svg">Illustration of the multisets for m=0..6</a>

%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 1

%e 3 6 3 1

%e 4 24 12 4 6 1

%e 5 120 60 20 30 5 10 1

%e 6 720 360 120 180 30 60 6 90 15 20 1

%Y Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

%K nonn,tabf

%O 0,3

%A _Tilman Piesk_, Nov 04 2014