A384904 - OEIS

A384904

Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

1, 1, 2, 3, 4, 5, 9, 9, 14, 17, 23, 25, 40, 41, 59, 68, 92, 99, 140, 151, 204, 229, 296, 328, 433, 476, 606, 685, 858, 955, 1203, 1336, 1654, 1858, 2266, 2537, 3102, 3453, 4169, 4680, 5611, 6262, 7495, 8358, 9927, 11105, 13096, 14613, 17227, 19179, 22459

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OFFSET

0,3

LINKS

John Tyler Rascoe, Table of n, a(n) for n = 0..100

FORMULA

G.f.: 1 + Sum_{i,k>0} q^((i*k*(2 + i*(k-1)))/2) * Product_{j=1..i-1} ( 1 + q^(2*k*j)/(1 - q^(k*j)) ) / (1 - q^(i*k)). - John Tyler Rascoe, Aug 20 2025

EXAMPLE

The partition (6,5,5,4,2,1) has maximal runs ((6,5),(5,4),(2,1)), with lengths (2,2,2), so is counted under a(23).

The a(1) = 1 through a(8) = 14 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (21) (22) (32) (33) (43) (44)

(111) (31) (41) (42) (52) (53)

(1111) (311) (51) (61) (62)

(11111) (222) (331) (71)

(321) (511) (422)

(411) (4111) (611)

(3111) (31111) (2222)

(111111) (1111111) (3221)

(3311)

(5111)

(41111)

(311111)

(11111111)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], SameQ@@Length/@Split[#, #2==#1-1&]&]], {n, 0, 30}]

PROG

(PARI) A_q(N) = {Vec(1+sum(k=1, floor(-1/2+sqrt(2+2*N)), sum(i=1, (N/(k*(k+1)/2))+1, q^((k*i*(2+i*(k-1)))/2)/(1-q^(k*i))*prod(j=1, i-1, 1 + q^(2*k*j)/(1 - q^(k*j))))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 20 2025

CROSSREFS

For subsets instead of strict partitions we have A243815, distinct lengths A384175.

For distinct instead of equal lengths we have A384882, counting gaps of 0 A384884.

The strict case is A384886.

Counting gaps of 0 gives A384887.

A000041 counts integer partitions, strict A000009.

A047993 counts partitions with max part = length (A106529).

A098859 counts Wilf partitions (complement A336866), compositions A242882.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

Cf. A000217, A008284, A047966, A089259, A325325, A382857, A383013, A383708, A384178, A384880.

Sequence in context: A182945 A389876 A360069 * A052270 A265335 A179223

Adjacent sequences: A384901 A384902 A384903 * A384905 A384906 A384907

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 20 2025

STATUS

approved