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A309992 Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows.

%I A309992 #32 Jan 27 2021 10:12:41
%S A309992 1,1,1,1,3,1,4,1,5,10,1,6,15,60,1,7,21,35,105,1,8,28,56,168,280,1,9,
%T A309992 36,84,126,252,504,1260,1,10,45,120,210,360,840,1260,2520,12600,1,11,
%U A309992 55,165,330,462,495,1320,2310,4620,6930,27720
%N A309992 Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows.
%C A309992 First row with repeated terms is row 15, see also A309999: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).
%H A309992 Alois P. Heinz, <a href="/A309992/b309992.txt">Rows n = 0..45, flattened</a>
%H A309992 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A309992 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A309992 For n = 5 there are 3 partitions of 5 into distinct parts: [5], [4,1], [3,2].  So row 5 contains M(5;5) = 1, M(5;4,1) = 5 and M(5;3,2) = 10.
%e A309992 Triangle T(n,k) begins:
%e A309992   1;
%e A309992   1;
%e A309992   1;
%e A309992   1,  3;
%e A309992   1,  4;
%e A309992   1,  5, 10;
%e A309992   1,  6, 15,  60;
%e A309992   1,  7, 21,  35, 105;
%e A309992   1,  8, 28,  56, 168, 280;
%e A309992   1,  9, 36,  84, 126, 252, 504, 1260;
%e A309992   1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;
%e A309992   1, 11, 55, 165, 330, 462, 495, 1320, 2310,  4620, 6930, 27720;
%e A309992   ...
%p A309992 g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, [], `if`(n=0, [1],
%p A309992      [map(x->binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))
%p A309992     end:
%p A309992 T:= n-> sort(g(n$2))[]:
%p A309992 seq(T(n), n=0..14);
%t A309992 g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[ Binomial[n, i] # & /@ g[n-i, Min[n-i, i-1]], g[n, i-1]]]];
%t A309992 T[n_] := Sort[g[n, n]];
%t A309992 T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Jan 27 2021, after _Alois P. Heinz_ *)
%Y A309992 Columns k=1-3 give: A000012, A000027 (for n>=3), A000217(n-1) (for n>=5).
%Y A309992 Row sums give A007837.
%Y A309992 Rightmost terms of rows give A290517.
%Y A309992 Cf. A000009, A036038, A309999, A325901, A325903.
%K A309992 nonn,tabf
%O A309992 0,5
%A A309992 _Alois P. Heinz_, Aug 26 2019