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

0,3

Comments

T(0,0) = 1 by convention.

Links

Alois P. Heinz, Rows n = 0..150, flattened

Wikipedia, Partition of a set

Example

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;
  ...

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 *)

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.

Sequence in context: A169891 A137756 A333468 * A225489 A308773 A051732

Adjacent sequences: A364968 A364969 A364970 * A364972 A364973 A364974

Keyword

nonn,tabf

Author

Alois P. Heinz, Aug 15 2023

Status

approved