cp's OEIS Frontend

A384904 Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

%I A384904 #15 Aug 21 2025 06:13:34
%S A384904 1,1,2,3,4,5,9,9,14,17,23,25,40,41,59,68,92,99,140,151,204,229,296,
%T A384904 328,433,476,606,685,858,955,1203,1336,1654,1858,2266,2537,3102,3453,
%U A384904 4169,4680,5611,6262,7495,8358,9927,11105,13096,14613,17227,19179,22459
%N A384904 Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.
%H A384904 John Tyler Rascoe, <a href="/A384904/b384904.txt">Table of n, a(n) for n = 0..100</a>
%F A384904 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 A384904 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 A384904 The a(1) = 1 through a(8) = 14 partitions:
%e A384904   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A384904        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A384904              (111)  (31)    (41)     (42)      (52)       (53)
%e A384904                     (1111)  (311)    (51)      (61)       (62)
%e A384904                             (11111)  (222)     (331)      (71)
%e A384904                                      (321)     (511)      (422)
%e A384904                                      (411)     (4111)     (611)
%e A384904                                      (3111)    (31111)    (2222)
%e A384904                                      (111111)  (1111111)  (3221)
%e A384904                                                           (3311)
%e A384904                                                           (5111)
%e A384904                                                           (41111)
%e A384904                                                           (311111)
%e A384904                                                           (11111111)
%t A384904 Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]
%o A384904 (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 A384904 For subsets instead of strict partitions we have A243815, distinct lengths A384175.
%Y A384904 For distinct instead of equal lengths we have A384882, counting gaps of 0 A384884.
%Y A384904 The strict case is A384886.
%Y A384904 Counting gaps of 0 gives A384887.
%Y A384904 A000041 counts integer partitions, strict A000009.
%Y A384904 A047993 counts partitions with max part = length (A106529).
%Y A384904 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A384904 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384904 Cf. A000217, A008284, A047966, A089259, A325325, A382857, A383013, A383708, A384178, A384880.
%K A384904 nonn
%O A384904 0,3
%A A384904 _Gus Wiseman_, Jun 20 2025