A367955 Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1

Offset: 0

Author

Alois P. Heinz, Dec 05 2023

Keywords

• nonn
• tabf

Comments

Rows and also columns reversed converge to A365441.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.

Examples

			T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  .  1;
  .  .  1, 1;
  .  .  .  1, 1, 2, 1;
  .  .  .  .  1, 1, 2, 5, 2,  3,  1;
  .  .  .  .  .  1, 1, 2, 5, 10,  7,  7, 11,  3,  4,  1;
  .  .  .  .  .  .  1, 1, 2,  5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Wikipedia, Partition of a set

Crossrefs

Row sums give A000110.

Column sums give A204856.

Antidiagonal sums give A368102.

T(2n,3n) gives A365441.

T(n,2n) gives A368675.

Row maxima give A367969.

Row n has A000124(n-1) terms (for n>=1).

Cf. A000217, A124327 (the same for block minima), A200660, A278677.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
T[0, 0] = 1; T[n_, k_] := b[k, n, 0];
Table[Table[T[n, k], {k, n, n*(n + 1)/2}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz's second Maple program *)

Formula

Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n-1) for n>=1.

Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.

T(n,n) = T(n,n*(n+1)/2) = 1.