A080510 -id:A080510 - cp's OEIS frontend

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

Alois P. Heinz, Aug 15 2023

T(0,0) = 1 by convention.

			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;
  ...

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

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.

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);

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 *)

A108947 Triangle: T(n,k) is the partition function G(n-k,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 2, 1, 1, 0, 1, 10, 5, 2, 1, 1, 0, 1, 26, 14, 5, 2, 1, 1, 0, 1, 76, 46, 15, 5, 2, 1, 1, 0, 1, 232, 166, 51, 15, 5, 2, 1, 1, 0, 1, 764, 652, 196, 52, 15, 5, 2, 1, 1, 0, 1, 2620, 2780, 827, 202, 52, 15, 5, 2, 1, 1

Offset: 0

Paul Boddington, Jul 21 2005

See entries for A001680 and A001681 for appropriate references.

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000110. First differences of a sequence G(k, 0), G(k, 1), ... give a row of A080510 (e.g., 0, 1, 10, 14, 15, 15, ... gives 1, 9, 4, 1).

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(G(n-i*j, i-1)*n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> G(n-k, k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1]*n!/i!^j/(n-i*j)! /j!, {j, 0, n/i}]]]; T[n_, k_] := G[n-k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

E.g.f. for sequence G(0, k), G(1, k), ... is exp(x + (1/2)*x^2 + ... + (1/k!)*x^k).

One term corrected by Alois P. Heinz, Sep 15 2013

cp's OEIS Frontend

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.

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