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A215520 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part <= k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A215520 #32 Aug 26 2019 17:50:22
%S A215520 1,1,2,1,2,3,1,3,4,5,1,3,5,6,7,1,4,7,9,10,11,1,4,8,10,12,13,14,1,5,10,
%T A215520 14,17,18,19,20,1,5,12,16,21,23,25,26,27,1,6,14,20,27,29,32,34,35,36,
%U A215520 1,6,16,22,32,35,40,43,45,46,47,1,7,19,28,40,45,52,57,60,62,63,64
%N A215520 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part <= k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%C A215520 Differs from A026820 first at T(7,4).
%C A215520 Differs from A213008 first at T(8,3).
%H A215520 Alois P. Heinz, <a href="/A215520/b215520.txt">Rows n = 1..75, flattened</a>
%H A215520 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%e A215520 T(3,2) =  2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
%e A215520 T(5,2) =  3 = |{30, 60, 120}|.
%e A215520 T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
%e A215520 T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
%e A215520 T(9,2) =  5 = |{22680, 45360, 90720, 181440, 362880}|.
%e A215520 Triangle T(n,k) begins:
%e A215520   1;
%e A215520   1,  2;
%e A215520   1,  2,  3;
%e A215520   1,  3,  4,  5;
%e A215520   1,  3,  5,  6,  7;
%e A215520   1,  4,  7,  9, 10, 11;
%e A215520   1,  4,  8, 10, 12, 13, 14;
%e A215520   1,  5, 10, 14, 17, 18, 19, 20;
%e A215520   1,  5, 12, 16, 21, 23, 25, 26, 27;
%e A215520   1,  6, 14, 20, 27, 29, 32, 34, 35, 36;
%p A215520 b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
%p A215520       {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
%p A215520     end:
%p A215520 T:= (n, k)-> nops(b(n, k)):
%p A215520 seq(seq(T(n, k), k=1..n), n=1..14);
%t A215520 b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jan 21 2015, after _Alois P. Heinz_ *)
%Y A215520 Main diagonal gives: A070289.
%Y A215520 Columns k=1-3 give: A000012, A008619 (for n>1), A001399 (for n>2).
%Y A215520 Cf. A026820, A213008, A215521.
%K A215520 nonn,tabl
%O A215520 1,3
%A A215520 _Alois P. Heinz_, Aug 14 2012