A375405 Number of integer partitions of n with a repeated part other than the least.

0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 13, 20, 29, 42, 62, 83, 117, 158, 214, 283, 377, 488, 641, 823, 1058, 1345, 1714, 2154, 2713, 3387, 4222, 5230, 6474, 7959, 9782, 11956, 14591, 17737, 21529, 26026, 31422, 37811, 45425, 54418, 65097, 77652, 92510, 109943, 130468

Offset

0,8

Comments

Also partitions whose minima of maximal anti-runs are not identical. An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from John Tyler Rascoe)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Formula

G.f.: Sum_{i>0} (Sum_{j>i} ( (x^(i+(2*j)))/(1-x^i) * Product_{k>=i} (1-[k<j]*x^(2*k))/(1-x^k) )). - John Tyler Rascoe, Aug 21 2024

Example

The a(0) = 0 through a(10) = 13 partitions:
  .  .  .  .  .  (221)  (2211)  (331)    (332)     (441)      (442)
                                (2221)   (3221)    (3321)     (3322)
                                (22111)  (3311)    (4221)     (3331)
                                         (22211)   (22221)    (4411)
                                         (221111)  (32211)    (5221)
                                                   (33111)    (32221)
                                                   (222111)   (33211)
                                                   (2211111)  (42211)
                                                              (222211)
                                                              (322111)
                                                              (331111)
                                                              (2221111)
                                                              (22111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !SameQ@@Min/@Split[#, UnsameQ]&]], {n, 0, 30}]
- or -
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@DeleteCases[#, Min@@#]&]], {n, 0, 30}]

Prog

(PARI)
A_x(N) = {my(x='x+O('x^N), f=sum(i=1, N, sum(j=i+1, N-i, ((x^(i+(2*j)))/(1-x^i))*prod(k=i+1, N-i-(2*j), if(k<j, 1-x^(2*k), 1)/(1-x^k))))); concat([0, 0, 0, 0, 0], Vec(f))}
A_x(60) \\ John Tyler Rascoe, Aug 21 2024

Crossrefs

The complement for maxima instead of minima is A034296.

The complement is counted by A115029, ranks A375396.

For maxima instead of minima we have A239955, ranks A073492.

These partitions have ranks A375397.

For distinct instead of identical we have A375404, ranks A375399.

A000041 counts integer partitions, strict A000009.

A003242 counts anti-run compositions, ranks A333489.

A055887 counts sequences of partitions with total sum n.

A375128 lists minima of maximal anti-runs of prime indices, sums A374706.

Cf. A046660, A065200, A065201, A141199, A358836, A374704, A375133, A375134, A375136, A375401.

Sequence in context: A035424 A245934 A099351 * A265065 A265066 A265067

Adjacent sequences: A375402 A375403 A375404 * A375406 A375407 A375408

Keyword

nonn

Author

Gus Wiseman, Aug 17 2024

Status

approved