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.
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, 14, 17, 18, 19, 20, 1, 5, 12, 16, 21, 23, 25, 26, 27, 1, 6, 14, 20, 27, 29, 32, 34, 35, 36, 1, 6, 16, 22, 32, 35, 40, 43, 45, 46, 47, 1, 7, 19, 28, 40, 45, 52, 57, 60, 62, 63, 64
Offset: 1
Examples
T(3,2) = 2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
T(5,2) = 3 = |{30, 60, 120}|.
T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
T(9,2) = 5 = |{22680, 45360, 90720, 181440, 362880}|.
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;
1, 5, 10, 14, 17, 18, 19, 20;
1, 5, 12, 16, 21, 23, 25, 26, 27;
1, 6, 14, 20, 27, 29, 32, 34, 35, 36;
Links
- Alois P. Heinz, Rows n = 1..75, flattened
- Wikipedia, Multinomial coefficients
Crossrefs
Programs
-
Maple
b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {}, {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)})) end: T:= (n, k)-> nops(b(n, k)): seq(seq(T(n, k), k=1..n), n=1..14); -
Mathematica
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 *)
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