A372650 -id:A372650 - cp's OEIS frontend

A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 31, 10, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 344, 336, 140, 35, 21, 0, 1, 0, 1661, 1393, 616, 385, 56, 28, 0, 1, 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1, 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1

Offset: 0

Alois P. Heinz, May 12 2024

			T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,4) = 0.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,    1;
  0,    31,    10,    10,    0,    1;
  0,    82,    70,    35,   15,    0,   1;
  0,   344,   336,   140,   35,   21,   0,   1;
  0,  1661,  1393,   616,  385,   56,  28,   0,  1;
  0,  7942,  6210,  4984, 1386,  504,  84,  36,  0, 1;
  0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-2 give: A000007, A224219, A372764.
Row sums give A000110.
T(2n,n) gives A271425.
Cf. A372650, A372722.

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

Sum_{k=0..n} k * T(n,k) = A372650(n).

A372649 Total sum over all partitions of [n] of the number of maximal blocks.

Original entry on oeis.org

0, 1, 3, 7, 21, 71, 293, 1268, 6107, 31123, 170745, 998966, 6212627, 40854360, 283290348, 2059884614, 15667307457, 124266461587, 1025342179759, 8784261413616, 78003593175261, 716854898767936, 6808817431686858, 66754426111124686, 674754718441688851

Offset: 0

Alois P. Heinz, May 08 2024

nonn

			a(3) = 7 = 3 + 1 + 1 + 1 + 1: 1|2|3, 1|23, 12|3, 13|2, 123.
a(4) = 21 = 1+1+1+2+1+1+2+1+2+1+1+1+1+1+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..576
Wikipedia, Partition of a set

Cf. A000110, A097148, A372650, A372722.

b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, max(j, m),
     `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..24);

b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
   Sum[Binomial[n - 1, j - 1]*b[n - j, Max[j, m],
   If[j > m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 10 2024, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} k * A372722(n,k).

cp's OEIS Frontend

A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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