A393921 Number of ordered set partitions of [n] whose set of block sizes covers an initial interval.

1, 1, 2, 12, 60, 450, 3960, 40320, 470400, 6161400, 89737200, 1433678400, 24985144800, 470929258800, 9552934591200, 207416225016000, 4800763199232000, 117985032861696000, 3068809378322688000, 84220901965565164800, 2432308571799698592000, 73740283173518978016000

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..424

Example

The ordered set partition of [6], 4|23|16|5 has the set of block sizes {1,2} so it is counted under a(6) = 3960.

Maple

b:= proc(n, i, p) option remember; uses combinat; `if`(n=0, p!, add(
      b(n-i*j, i+1, p+j)*multinomial(n, n-i*j, i$j)/j!, j=1..n/i))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 03 2026

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, Sum[b[n-i*j, i+1, p+j]*Multinomial[n-i*j, Sequence @@ Table[i, {j}]]/j!, {j, 1, n/i}]];
a[n_] := b[n, 1, 0];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, May 08 2026, after Alois P. Heinz *)

Prog

(PARI)
subsets(S) = {my(s=List()); for(i=0, 2^(#S) -1, my(x=List()); for(j=1, #S, if(bitand(i, 1<<(j-1)), listput(x, S[j]))); listput(s, Vec(x))); Vec(s)}
C_x(k, N) = { my(x='x+O('x^(N+1)), S = subsets(vector(k, i, i)),  v= sum(i=2, #S, (-1)^(k-#S[i])/(1-sum(k=1, #S[i], x^S[i][k] / S[i][k]!)))); v}
D_x(N) = { my(x='x+O('x^(N+1))); Vec(serlaplace(1 + sum(i=1, floor(sqrt(2*N)+1/2), C_x(i, N))))}

Crossrefs

Cf. A000670, A393098.

Sequence in context: A360590 A386497 A362244 * A362238 A389817 A372986

Adjacent sequences: A393918 A393919 A393920 * A393922 A393923 A393924

Keyword

nonn

Author

John Tyler Rascoe, Mar 02 2026

Status

approved