A355388 Number of composable pairs (y, v) of integer compositions of n, where a composition is regarded as an arrow from the number of parts to the number of distinct parts.

1, 1, 2, 6, 18, 58, 174, 536, 1656, 4947, 14800, 43157, 126572, 364070, 1039926, 2938898, 8223400, 22846370, 62930113, 172177400, 467002792, 1259736804, 3371190792, 8973530491, 23728305128, 62421018163, 163255839779, 424842462529, 1100006243934, 2834558927244, 7270915592897

Offset: 0

Author

Gus Wiseman, Jul 02 2022

Keywords

• nonn

Comments

Being composable here means that the length of v equals the number of distinct parts in y.

Examples

			The a(0) = 1 through a(4) = 18 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(2)  (21)(21)  (31)(31)
                         (21)(12)  (31)(13)
                         (12)(21)  (31)(22)
                         (12)(12)  (13)(31)
                         (111)(3)  (13)(13)
                                   (13)(22)
                                   (22)(4)
                                   (211)(31)
                                   (211)(13)
                                   (211)(22)
                                   (121)(31)
                                   (121)(13)
                                   (121)(22)
                                   (112)(31)
                                   (112)(13)
                                   (112)(22)
                                   (1111)(4)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800 (first 201 terms from Andrew Howroyd)

Crossrefs

The case with containment is A032020.

The inhomogeneous version with containment is A355384, partitions A355383.

The version for partitions is A355385, with containment A000009.

A133494 counts compositions of each part of a composition, strict A336139.

A323583 counts splittings of partitions.

Cf. A001970, A022811, A063834, A070933, A316245, A319910, A355382.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n-1, i-1), i=0..degree(p)))(b(n$2, 0)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 01 2023

Mathematica

Table[Length[Select[Tuples[Join@@Permutations/@IntegerPartitions[n],2], Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,10}]

PARI

a(n) = {if(n==0, 1, my(s=0); forpart(p=n, p=Vec(p); my(S=Set(p)); s += binomial(n-1, #S-1)*(#p)!/prod(i=1, #S, my(c=#select(t->t==S[i], p)); c! )); s)} \\ Andrew Howroyd, Jan 01 2023

PARI

\\ for larger n.
a(n) = { local(Cache=Map());
  my(F(r,m,p,q) = my(key=[r,m,p,q], z); if(!mapisdefined(Cache, key, &z),
  z = if(m==0, if(r==0, p!*binomial(n-1, q-1)), self()(r, m-1, p, q) + sum(j=1, r\m, self()(r-j*m, min(m-1, r-j*m), p+j, q+1)/j!));
  mapput(Cache, key, z) ); z);
  if(n==0, 1, F(n, n, 0, 0))
} \\ Andrew Howroyd, Jan 01 2023

Formula

a(n) = Sum_{k>=1} binomial(n-1, k-1)*A235998(n, k) for n > 0. - Andrew Howroyd, Jan 01 2023

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 01 2023