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

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, 14, 16, 18, 19, 20, 1, 5, 12, 17, 21, 23, 25, 26, 27, 1, 6, 13, 21, 26, 30, 32, 34, 35, 36, 1, 6, 16, 25, 33, 37, 41, 43, 45, 46, 47, 1, 7, 19, 32, 42, 50, 54, 58, 60, 62, 63, 64

Offset

1,3

Comments

Differs from A026820 after position 24.

Includes sequence A070289 when k = n.

Links

Alois P. Heinz, Rows n = 1..45, flattened

Katsuhisa Yamanaka, Shin-ichiro Kawano, Yosuke Kikuchi, and Shin-ichi Nakano, Constant Time Generation of Integer Partitions, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E90-A, no.5, pp. 888-895, (May-2007).

Sergei Viznyuk, C-Program for this sequence, 2012.

Sergei Viznyuk, C-Program for sequences A026820, A070289, and A213008 (local copy), 2012.

Example

Triangle T(n,k) begins:
  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;
  ...
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!).

Maple

b:= proc(n, i, k) option remember; if n=0 then {1} elif i<1
      then {} else {b(n, i-1, k)[], seq(map(x-> x*i!^j,
              b(n-i*j, i-1, k-j))[], j=1..min(n/i, k))} fi
    end:
T:= (n, k)-> nops(b(n, n, k)):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 14 2012

Mathematica

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 *)

Crossrefs

Cf. A026820, A070289.

Sequence in context: A066010 A209556 A109974 * A215520 A026820 A330661

Adjacent sequences: A213005 A213006 A213007 * A213009 A213010 A213011

Keyword

nonn,tabl

Author

Sergei Viznyuk, Jun 01 2012

Status

approved