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

0,4

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

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

Example

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;
  ...

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 *)

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.

Sequence in context: A153861 A118981 A117938 * A101912 A208522 A209569

Adjacent sequences: A256190 A256191 A256192 * A256194 A256195 A256196

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 19 2015

Status

approved