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
Examples
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!).
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.
Programs
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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 *)
Comments