A364971 Number T(n,k) of partitions of [n] for which the difference between the longest and the shortest block size is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

1, 1, 2, 2, 3, 5, 6, 4, 2, 35, 10, 5, 27, 60, 95, 15, 6, 2, 371, 315, 161, 21, 7, 142, 938, 2002, 770, 252, 28, 8, 282, 4005, 9744, 5313, 1386, 372, 36, 9, 1073, 16950, 50275, 33705, 11082, 2310, 525, 45, 10, 2, 74657, 283525, 217800, 78078, 20097, 3630, 715, 55, 11

Offset: 0

Author

Alois P. Heinz, Aug 15 2023

Keywords

• nonn
• tabf

Comments

T(0,0) = 1 by convention.

Examples

			T(4,0) = 5: 1|2|3|4, 12|34, 13|24, 14|23, 1234.
T(4,1) = 6: 1|2|34, 1|23|4, 1|24|3, 12|3|4, 13|2|4, 14|2|3.
T(4,2) = 4: 1|234, 123|4, 124|3, 134|2.
Triangle T(n,k) begins:
     1;
     1;
     2;
     2,     3;
     5,     6,     4;
     2,    35,    10,     5;
    27,    60,    95,    15,     6;
     2,   371,   315,   161,    21,    7;
   142,   938,  2002,   770,   252,   28,   8;
   282,  4005,  9744,  5313,  1386,  372,  36,  9;
  1073, 16950, 50275, 33705, 11082, 2310, 525, 45, 10;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened
• Wikipedia, Partition of a set

Crossrefs

Row sums give A000110.

Column k=0 gives A038041 (for n>=1).

T(n,n-2) gives A000027 (for n>=2).

Cf. A080510, A178979, A364967.

Programs

Maple

b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
     b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..12);

Mathematica

b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := CoefficientList[b[n, n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)