A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107

Offset

0,4

Comments

An anti-run is a sequence with no adjacent equal parts.

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

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

Links

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

Formula

G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (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 maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
                (21)  (31)   (32)   (42)   (43)    (53)    (54)
                      (211)  (41)   (51)   (52)    (62)    (63)
                             (311)  (321)  (61)    (71)    (72)
                                    (411)  (322)   (422)   (81)
                                           (421)   (431)   (432)
                                           (511)   (521)   (522)
                                           (3211)  (611)   (531)
                                                   (3221)  (621)
                                                   (4211)  (711)
                                                           (4221)
                                                           (4311)
                                                           (5211)
                                                           (32211)

Mathematica

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

Prog

(PARI)
A_x(N) = {my(x='x+O('x^N), f=sum(i=0, N, (x^i)*prod(j=1, i-1, (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 see: A034296, A115029, A374760, A374759.

For compositions instead of partitions we have A374761.

For minima instead of maxima we have A375134, ranks A375398.

The complement is counted by A375401, ranks A375403.

These partitions are ranked by A375402, for compositions A374767.

The complement for minima instead of maxima is 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. A141199, A279790, A358830, A358833, A358836, A358905, A374704, A374757, A374758, A375136, A375400.

Sequence in context: A080713 A058664 A238479 * A035562 A107234 A035943

Adjacent sequences: A375130 A375131 A375132 * A375134 A375135 A375136

Keyword

nonn

Author

Gus Wiseman, Aug 14 2024

Status

approved