A274174 Number of compositions of n if all summand runs are kept together.

1, 1, 2, 4, 7, 12, 22, 36, 60, 97, 162, 254, 406, 628, 974, 1514, 2305, 3492, 5254, 7842, 11598, 17292, 25294, 37090, 53866, 78113, 112224, 161092, 230788, 328352, 466040, 658846, 928132, 1302290, 1821770, 2537156, 3536445, 4897310, 6777806, 9341456, 12858960, 17625970, 24133832, 32910898, 44813228, 60922160, 82569722

Offset: 0

Author

Gregory L. Simay, Jun 12 2016

Keywords

• nonn

Comments

a(n^2) is odd. - Gregory L. Simay, Jun 23 2019

Also the number of compositions of n avoiding the patterns (1,2,1) and (2,1,2). - Gus Wiseman, Jul 07 2020

Examples

			If the summand runs are blocked together, there are 22 compositions of a(6): 6; 5+1, 1+5, 4+2, 2+4, (3+3), 4+(1+1), (1+1)+4, 1+2+3, 1+3+2, 2+1+3, 2+3+1, 3+1+2, 3+2+1, (2+2+2), 3+(1+1+1), (1+1+1)+3, (2+2)+(1+1), (1+1)+(2+2), 2+(1+1+1+1), (1+1+1+1)+2, (1+1+1+1+1+1).
a(0)=1; a(1)= 1; a(4) = 7; a(9) = 97; a(16) = 2305; a(25) = 78113 and a(36) = 3536445. - _Gregory L. Simay_, Jun 23 2019
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

Cf. A000070, A101880, A116608.

The version for patterns is A001339.

The version for prime indices is A333175.

The complement (i.e., the matching version) is A335548.

Anti-run compositions are A003242.

(1,2,1)- and (2,1,2)-matching permutations of prime indices are A335462.

(1,2,1)-matching compositions are A335470.

(1,2,1)-avoiding compositions are A335471.

(2,1,2)-matching compositions are A335472.

(2,1,2)-avoiding compositions are A335473.

Cf. A000670, A056986, A181796, A335451, A335452, A335460, A335463.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       add(b(n-i*j, i-1, p+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 12 2016

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==Length[Union[#]]&]],{n,0,10}] (* Gus Wiseman, Jul 07 2020 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
    Sum[b[n - i*j, i - 1, p + If[j == 0, 0, 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 11 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=0} k! * A116608(n,k). - Joerg Arndt, Jun 12 2016

Extensions

Terms a(9) and beyond from Joerg Arndt, Jun 12 2016