A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075

Offset: 0

Author

Gus Wiseman, Aug 04 2024

Keywords

• nonn

Examples

			The a(0) = 1 through a(4) = 15 ways:
  ()  ((1))  ((2))      ((3))          ((4))
             ((1,1))    ((1,2))        ((1,3))
             ((1),(1))  ((1,1,1))      ((2,2))
                        ((1),(1,1))    ((1,1,2))
                        ((1,1),(1))    ((2),(2))
                        ((1),(1),(1))  ((1,1,1,1))
                                       ((1),(1,2))
                                       ((1,2),(1))
                                       ((1),(1,1,1))
                                       ((1,1),(1,1))
                                       ((1,1,1),(1))
                                       ((1),(1),(1,1))
                                       ((1),(1,1),(1))
                                       ((1,1),(1),(1))
                                       ((1),(1),(1),(1))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

A variation for weakly increasing lengths is A141199.

For identical sums instead of minima we have A279787.

The case of reversed twice-partitions is A306319, distinct A358830.

For maxima instead of minima, or for unreversed partitions, we have A358905.

The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

A055887 counts sequences of partitions with total sum n.

A281145 counts same-trees.

A319169 counts partitions with constant Omega, ranked by A320324.

A358911 counts compositions with constant Omega, distinct A358912.

Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.

Programs

Mathematica

Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]

PARI

seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024

Formula

G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - Andrew Howroyd, Dec 29 2024

Extensions

a(16) onwards from Andrew Howroyd, Dec 29 2024