A375134 Number of integer partitions of n whose maximal anti-runs have distinct minima.

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 18, 21, 28, 33, 43, 52, 66, 78, 98, 116, 145, 171, 209, 247, 300, 352, 424, 499, 595, 695, 826, 963, 1138, 1322, 1553, 1802, 2106, 2435, 2835, 3271, 3795, 4365, 5046, 5792, 6673, 7641, 8778, 10030, 11490, 13099, 14968, 17030

Offset

0,4

Comments

These are partitions with no part appearing more than twice and with the least part appearing only once.

Also the number of reversed integer partitions of n whose maximal anti-runs have distinct minima.

Links

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Formula

G.f.: 1 + Sum_{i>0} (x^i * Product_{j>i} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

Example

The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with minima (5,3,1), so y is counted under a(29).
The a(1) = 1 through a(9) = 11 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (12)  (13)  (14)   (15)   (16)   (17)    (18)
                        (23)   (24)   (25)   (26)    (27)
                        (122)  (123)  (34)   (35)    (36)
                                      (124)  (125)   (45)
                                      (133)  (134)   (126)
                                             (233)   (135)
                                             (1223)  (144)
                                                     (234)
                                                     (1224)
                                                     (1233)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Min/@Split[#, UnsameQ]&]], {n, 0, 30}]

Prog

(PARI)
A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1, N, (x^i)*prod(j=i+1, N-i, (1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024

Crossrefs

Includes all strict partitions A000009.

For identical instead of distinct leaders we have A115029.

A version for compositions instead of partitions is A374518, ranks A374638.

For minima instead of maxima we have A375133, ranks A375402.

These partitions have ranks A375398.

The complement is counted by A375404, ranks A375399.

A000041 counts integer partitions.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts integer compositions.

A055887 counts sequences of partitions with total sum n.

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

Cf. A034296, A141199, A358830, A358836, A358905, A374704, A374757, A374758, A374761, A375136, A375400, A375401.

Sequence in context: A227135 A162417 A240012 * A295261 A293627 A264393

Adjacent sequences: A375131 A375132 A375133 * A375135 A375136 A375137

Keyword

nonn

Author

Gus Wiseman, Aug 14 2024

Status

approved