This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384886 #12 Aug 21 2025 06:14:07
%S A384886 1,1,1,2,2,3,4,4,4,7,7,8,11,11,14,17,19,20,27,27,35,38,45,47,60,63,75,
%T A384886 84,97,104,127,134,155,175,196,218,251,272,307,346,384,424,480,526,
%U A384886 586,658,719,798,890,979,1078,1201,1315,1451,1603,1762,1934,2137
%N A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).
%H A384886 John Tyler Rascoe, <a href="/A384886/b384886.txt">Table of n, a(n) for n = 0..100</a>
%F A384886 G.f.: 1 + Sum_{i,k>0} q^(k*(k+1)*i^2/2)/Product_{j=1..i} (1 - q^(j*k)). - _John Tyler Rascoe_, Aug 21 2025
%e A384886 The strict partition y = (7,6,5,3,2,1) has maximal runs ((7,6,5),(3,2,1)), with lengths (3,3), so y is counted under a(24).
%e A384886 The a(1) = 1 through a(14) = 14 partitions (A-E = 10-14):
%e A384886 1 2 3 4 5 6 7 8 9 A B C D E
%e A384886 21 31 32 42 43 53 54 64 65 75 76 86
%e A384886 41 51 52 62 63 73 74 84 85 95
%e A384886 321 61 71 72 82 83 93 94 A4
%e A384886 81 91 92 A2 A3 B3
%e A384886 432 631 A1 B1 B2 C2
%e A384886 531 4321 641 543 C1 D1
%e A384886 731 642 742 752
%e A384886 741 751 842
%e A384886 831 841 851
%e A384886 5421 931 941
%e A384886 A31
%e A384886 5432
%e A384886 6521
%t A384886 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,15}]
%o A384886 (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*(k+1)*i^2/2)/prod(j=1,i, 1 - q^(j*k)))) + O('q^(N+1)))} \\ _John Tyler Rascoe_, Aug 21 2025
%Y A384886 For subsets instead of strict partitions we have A243815, distinct lengths A384175.
%Y A384886 For distinct instead of equal lengths we have A384178, for anti-runs A384880.
%Y A384886 This is the strict case of A384904, distinct lengths A384884.
%Y A384886 A000041 counts integer partitions, strict A000009.
%Y A384886 A047993 counts partitions with max part = length (A106529).
%Y A384886 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A384886 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384886 Cf. A000217, A008284, A044813, A047966, A089259, A325324, A325325, A329739, A382857, A383013, A383708, A384176.
%K A384886 nonn
%O A384886 0,4
%A A384886 _Gus Wiseman_, Jun 13 2025