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A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

%I A249619 #15 Nov 01 2021 03:15:28
%S A249619 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 A249619 6,90,15,20,1,5040,2520,840,1260,210,420,42,630,105,140,7,210,21,35,1,
%U A249619 40320,20160,6720,10080,1680,3360,336,5040,840,1120,56
%N A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).
%C A249619 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 A249619 Row lengths: A000041 (partition numbers), Row sums: A005651
%C A249619 Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
%C A249619 Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
%C A249619 The rightmost columns form the triangle A173333:
%C A249619   n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
%C A249619 m
%C A249619 1         1
%C A249619 2         2      1
%C A249619 3         6      3     1
%C A249619 4        24     12     4     1
%C A249619 5       120     60    20     5    1
%C A249619 6       720    360   120    30    6   1
%C A249619 7      5040   2520   840   210   42   7   1
%C A249619 8     40320  20160  6720  1680  336  56   8   1
%C A249619 A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.
%H A249619 Tilman Piesk, <a href="/A249619/b249619.txt">Triangle rows m=0..25, flattened</a>
%H A249619 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 A249619 Triangle begins:
%e A249619   n     0    1    2    3   4   5  6   7   8   9 10
%e A249619 m
%e A249619 0       1
%e A249619 1       1
%e A249619 2       2    1
%e A249619 3       6    3    1
%e A249619 4      24   12    4    6   1
%e A249619 5     120   60   20   30   5  10  1
%e A249619 6     720  360  120  180  30  60  6  90  15  20  1
%Y A249619 Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.
%K A249619 nonn,tabf
%O A249619 0,3
%A A249619 _Tilman Piesk_, Nov 04 2014