A256193 Number T(n,k) of partitions of n into two sorts of parts having exactly k parts of the second sort; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 5, 12, 11, 5, 1, 7, 20, 24, 16, 6, 1, 11, 35, 49, 41, 22, 7, 1, 15, 54, 89, 91, 63, 29, 8, 1, 22, 86, 158, 186, 155, 92, 37, 9, 1, 30, 128, 262, 351, 342, 247, 129, 46, 10, 1, 42, 192, 428, 635, 700, 590, 376, 175, 56, 11, 1

Offset: 0

Author

Alois P. Heinz, Mar 19 2015

Keywords

• nonn
• tabl

Examples

			T(3,0) = 3: 111, 21, 3.
T(3,1) = 6: 1'11, 11'1, 111', 2'1, 21', 3'.
T(3,2) = 4: 1'1'1, 1'11', 11'1', 2'1'.
T(3,3) = 1: 1'1'1'.
Triangle T(n,k) begins:
   1;
   1,   1;
   2,   3,   1;
   3,   6,   4,   1;
   5,  12,  11,   5,   1;
   7,  20,  24,  16,   6,   1;
  11,  35,  49,  41,  22,   7,   1;
  15,  54,  89,  91,  63,  29,   8,   1;
  22,  86, 158, 186, 155,  92,  37,   9,  1;
  30, 128, 262, 351, 342, 247, 129,  46, 10,  1;
  42, 192, 428, 635, 700, 590, 376, 175, 56, 11,  1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• William Dugan, Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017.
• Emmy Huang and Ray Tang, Minimum Decomposition on Maxmin Trees, arXiv:2310.14385 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
• Wikipedia, q-Pochhammer symbol

Crossrefs

Column k=0-10 gives: A000041, A006128, A258472, A258473, A258474, A258475, A258476, A258477, A258478, A258479, A258480.

T(2n,n) gives A258471.

Row sums give A070933.

Cf. A278464.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
       binomial(j, t), t=0..j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]* Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]]; T[n_] := Function[ p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
Table[SeriesCoefficient[FunctionExpand[1/QPochhammer[q + x, q, n]], {q, 0, n - k}, {x, 0, k}], {n, 0, 10}, {k, 0, n}] // Column (* Vladimir Reshetnikov, Nov 22 2016 *)

Formula

T(n,k) = [x^k] [q^(n-k)] 1/(q+x; q)inf = [x^k] [q^(n-k)] 1/(q+x; q)_n, where (x; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016

Sum_{k=0..n} k * T(n,k) = A278464(n). - Alois P. Heinz, Nov 22 2016