login
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.
5

%I #32 Jan 27 2021 10:12:41

%S 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 36,84,126,252,504,1260,1,10,45,120,210,360,840,1260,2520,12600,1,11,

%U 55,165,330,462,495,1320,2310,4620,6930,27720

%N 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 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 Alois P. Heinz, <a href="/A309992/b309992.txt">Rows n = 0..45, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%e 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 Triangle T(n,k) begins:

%e 1;

%e 1;

%e 1;

%e 1, 3;

%e 1, 4;

%e 1, 5, 10;

%e 1, 6, 15, 60;

%e 1, 7, 21, 35, 105;

%e 1, 8, 28, 56, 168, 280;

%e 1, 9, 36, 84, 126, 252, 504, 1260;

%e 1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;

%e 1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720;

%e ...

%p g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, [], `if`(n=0, [1],

%p [map(x->binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))

%p end:

%p T:= n-> sort(g(n$2))[]:

%p seq(T(n), n=0..14);

%t 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 T[n_] := Sort[g[n, n]];

%t T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Jan 27 2021, after _Alois P. Heinz_ *)

%Y Columns k=1-3 give: A000012, A000027 (for n>=3), A000217(n-1) (for n>=5).

%Y Row sums give A007837.

%Y Rightmost terms of rows give A290517.

%Y Cf. A000009, A036038, A309999, A325901, A325903.

%K nonn,tabf

%O 0,5

%A _Alois P. Heinz_, Aug 26 2019