A372762 -id:A372762 - cp's OEIS frontend

A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.

1, 1, 4, 5, 31, 82, 344, 1661, 7942, 38721, 228680, 1377026, 8529756, 56756260, 402300799, 2960135917, 22692746719, 181667760724, 1516381486766, 13135566948285, 117868982320877, 1093961278908818, 10492653292100919, 103880022098900234, 1059925027073166856

Offset: 1

Geoffrey Critzer, Apr 01 2013

nonn

In other words, if the smallest block in a partition has size k then there are no other blocks in the partition with size k.

			a(4) = 5 because we have: {{1,2,3,4}}, {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}.

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A224244, A372721, A372802.

Column k=1 of A372762.

with(combinat):
b:= proc(n, i) option remember;
      `if`(i<1, 0, `if`(n=i, 1, 0)+add(b(n-i*j, i-1)*
       multinomial(n, n-i*j, i$j)/j!, j=0..(n-1)/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 07 2016

nn=25;Drop[Range[0,nn]!CoefficientList[Series[Sum[x^k/k!Exp[Exp[x]-Sum[x^i/i!,{i,0,k}]],{k,1,nn}],{x,0,nn}],x],1]
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[i<1, 0, If[n==i, 1, 0] + Sum[b[n-i*j, i-1]*multinomial[n, Prepend[Array[i&, j], n-i*j]]/j!, {j, 0, (n-1)/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)

E.g.f.: Sum_{k>=1} x^k/k! * exp(exp(x) - Sum_{i=0..k} x^i/i!).

A372722 Number T(n,k) of partitions of [n] having exactly k blocks of maximal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 36, 15, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 596, 175, 105, 0, 0, 0, 1, 0, 2809, 805, 420, 105, 0, 0, 0, 1, 0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1, 0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1, 0, 461748, 151371, 37730, 17325, 10395, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, May 11 2024

			T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,2) = 3: 12|34, 13|24, 14|23.
T(4,3) = 0.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,    1;
  0,    11,     3,    0,    1;
  0,    36,    15,    0,    0,   1;
  0,   132,    55,   15,    0,   0, 1;
  0,   596,   175,  105,    0,   0, 0, 1;
  0,  2809,   805,  420,  105,   0, 0, 0, 1;
  0, 14608,  4053, 1540,  945,   0, 0, 0, 0, 1;
  0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1;
  ...

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

Columns k=0-1 give: A000007, A372721.
Row sums give A000110.
T(2n,n) gives A001147.
T(3n,n) gives A271715.
Cf. A372649, A372762.

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

Sum_{k=0..n} k * T(n,k) = A372649(n).

A271425 Number of set partitions of [2n] with maximal block length multiplicity equal to n.

Original entry on oeis.org

1, 1, 9, 35, 385, 3717, 48279, 691119, 11229075, 200982925, 3928974907, 83060120871, 1885501840677, 45694145548625, 1176704027583075, 32077561625780175, 922854842240358825, 27951355368760441365, 889580295850449177975, 29707539555680924142975

Offset: 0

Alois P. Heinz, Apr 07 2016

nonn

In each set partition of [2n] counted by a(n) at least one block length occurs exactly n times, and all block lengths occur at most n times.

			a(1) = 1: 12.
a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(3) = 35: 123|4|5|6, 124|3|5|6, 12|34|56, 125|3|4|6, 12|35|46, 12|36|45, 126|3|4|5, 134|2|5|6, 13|24|56, 135|2|4|6, 13|25|46, 13|26|45, 136|2|4|5, 14|23|56, 1|234|5|6, 15|23|46, 1|235|4|6, 16|23|45, 1|236|4|5, 145|2|3|6, 14|25|36, 14|26|35, 146|2|3|5, 15|24|36, 1|245|3|6, 16|24|35, 1|246|3|5, 15|26|34, 16|25|34, 1|2|345|6, 1|2|346|5, 156|2|3|4, 1|256|3|4, 1|2|356|4, 1|2|3|456.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Cf. A271423, A372762.

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n$2, n)-b(2*n$2, n-1)):
seq(a(n), n=0..20);

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]*b[n - i*j, i-1, k]/j!, {j, 0, Min[k, n/i]}]]]; a[n_] := If[n==0, 1, b[2n, 2n, n] - b[2n, 2n, n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

a(n) = A271423(2n,n).

a(n) = A372762(2n,n). - Alois P. Heinz, May 12 2024

A372650 Total sum over all partitions of [n] of the number of minimal blocks.

Original entry on oeis.org

0, 1, 3, 7, 27, 86, 393, 1688, 8291, 44143, 248428, 1480073, 9440049, 63265606, 444309232, 3273807272, 25227429123, 202458174614, 1689474026499, 14636685675142, 131413462612012, 1220636654904548, 11712836883408675, 115956109213769404, 1182944504931376337

Offset: 0

Alois P. Heinz, May 08 2024

nonn

			a(4) = 27 = 1+1+1+2+2+1+2+2+2+1+2+2+2+2+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, A097147, A372649, A372762.

b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j b(n$2, 0):
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, Min[j, m],
   If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 11 2024, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} k * A372762(n,k).

A372764 Number of partitions of [n] having exactly two blocks of minimal size.

Original entry on oeis.org

0, 0, 1, 0, 9, 10, 70, 336, 1393, 6210, 41331, 228635, 1315974, 8779134, 61675419, 434566510, 3237964993, 25386526258, 207569429548, 1756564362651, 15418550267179, 140015129879331, 1316198207272686, 12786566843038549, 128035136707876270, 1319513338177755510

Offset: 0

Alois P. Heinz, May 12 2024

nonn

			a(2) = 1: 1|2.
a(4) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 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.

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

Column k=2 of A372762.

b:= proc(n, m, t) option remember; `if`(n=0,
     `if`(t=2, 1, 0), add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j b(n$2, 0):
seq(a(n), n=0..25);

cp's OEIS Frontend

A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.

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A372722 Number T(n,k) of partitions of [n] having exactly k blocks of maximal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A271425 Number of set partitions of [2n] with maximal block length multiplicity equal to n.

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A372650 Total sum over all partitions of [n] of the number of minimal blocks.

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Maple

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A372764 Number of partitions of [n] having exactly two blocks of minimal size.

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Maple