%I #15 Aug 21 2025 06:13:34

%S 1,1,2,3,4,5,9,9,14,17,23,25,40,41,59,68,92,99,140,151,204,229,296,

%T 328,433,476,606,685,858,955,1203,1336,1654,1858,2266,2537,3102,3453,

%U 4169,4680,5611,6262,7495,8358,9927,11105,13096,14613,17227,19179,22459

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

%H John Tyler Rascoe, <a href="/A384904/b384904.txt">Table of n, a(n) for n = 0..100</a>

%F 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

%e 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).

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

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

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

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

%e (1111) (311) (51) (61) (62)

%e (11111) (222) (331) (71)

%e (321) (511) (422)

%e (411) (4111) (611)

%e (3111) (31111) (2222)

%e (111111) (1111111) (3221)

%e (3311)

%e (5111)

%e (41111)

%e (311111)

%e (11111111)

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

%o (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

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

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

%Y The strict case is A384886.

%Y Counting gaps of 0 gives A384887.

%Y A000041 counts integer partitions, strict A000009.

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

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

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jun 20 2025