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

Author

Geoffrey Critzer, Apr 01 2013

Keywords

• nonn

Comments

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.

Examples

			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}}.
		

Links

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

Crossrefs

Cf. A224244, A372721, A372802.

Column k=1 of A372762.

Programs

Maple

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

Mathematica

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 *)

Formula

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