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A213008 Triangle T(n,k) of number of distinct values of multinomial coefficients corresponding to sequence A026820 (n >= 1, 1 <= k <= n).

%I A213008 #43 May 24 2020 14:58:20
%S A213008 1,1,2,1,2,3,1,3,4,5,1,3,5,6,7,1,4,7,9,10,11,1,4,8,10,12,13,14,1,5,9,
%T A213008 14,16,18,19,20,1,5,12,17,21,23,25,26,27,1,6,13,21,26,30,32,34,35,36,
%U A213008 1,6,16,25,33,37,41,43,45,46,47,1,7,19,32,42,50,54,58,60,62,63,64
%N A213008 Triangle T(n,k) of number of distinct values of multinomial coefficients corresponding to sequence A026820 (n >= 1, 1 <= k <= n).
%C A213008 Differs from A026820 after position 24.
%C A213008 Includes sequence A070289 when k = n.
%H A213008 Alois P. Heinz, <a href="/A213008/b213008.txt">Rows n = 1..45, flattened</a>
%H A213008 Katsuhisa Yamanaka, Shin-ichiro Kawano, Yosuke Kikuchi, and Shin-ichi Nakano, <a href="https://doi.org/10.1093/ietfec/e90-a.5.888">Constant Time Generation of Integer Partitions</a>, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E90-A, no.5, pp. 888-895, (May-2007).
%H A213008 Sergei Viznyuk, <a href="http://phystech.com/ftp/A213008.c">C-Program for this sequence</a>, 2012.
%H A213008 Sergei Viznyuk, <a href="/A213008/a213008.c.txt">C-Program for sequences A026820, A070289, and A213008 (local copy)</a>, 2012.
%e A213008 Triangle T(n,k) begins:
%e A213008   1;
%e A213008   1, 2;
%e A213008   1, 2, 3;
%e A213008   1, 3, 4,  5;
%e A213008   1, 3, 5,  6,  7;
%e A213008   1, 4, 7,  9, 10, 11;
%e A213008   1, 4, 8, 10, 12, 13, 14;
%e A213008   ...
%e A213008 Thus, for n = 7 and k = 6 there are 13 distinct values of multinomial coefficients corresponding to partitions of n = 7 into at most k = 6 parts. The corresponding number of partitions from sequence A026820 is 14. That is because partitions 7 = 4 + 1 + 1 + 1 and 7 = 3 + 2 + 2 produce the same value of multinomial coefficient 7!/(4!*1!*1!*1!) = 7!/(3!*2!*2!).
%p A213008 b:= proc(n, i, k) option remember; if n=0 then {1} elif i<1
%p A213008       then {} else {b(n, i-1, k)[], seq(map(x-> x*i!^j,
%p A213008               b(n-i*j, i-1, k-j))[], j=1..min(n/i, k))} fi
%p A213008     end:
%p A213008 T:= (n, k)-> nops(b(n, n, k)):
%p A213008 seq(seq(T(n,k), k=1..n), n=1..14);  # _Alois P. Heinz_, Aug 14 2012
%t A213008 b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1}, If[i<1, {}, Join[b[n, i-1, k], Table[ Function[#*i!^j] /@ b[n-i*j, i-1, k-j], {j, 1, Min[n/i, k]}] // Flatten] // Union] ]; T[n_, k_] := Length[b[n, n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Mar 12 2015, after _Alois P. Heinz_ *)
%Y A213008 Cf. A026820, A070289.
%K A213008 nonn,tabl
%O A213008 1,3
%A A213008 _Sergei Viznyuk_, Jun 01 2012