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 A384178 #9 Jun 14 2025 23:51:08
%S A384178 1,1,1,2,1,2,2,3,3,4,5,6,6,8,8,10,11,13,13,16,15,19,19,23,22,26,28,31,
%T A384178 35,39,37,47,51,52,60,65,67,78,85,86,99,108,110,127,136,138,159,170,
%U A384178 171,196,209,213,240,257,260,292,306,313,350,371,369,417,441
%N A384178 Number of strict integer partitions of n with all distinct lengths of maximal runs (decreasing by 1).
%e A384178 The strict partition y = (9,7,6,5,2,1) has maximal runs ((9),(7,6,5),(2,1)), with lengths (1,3,2), so y is counted under a(30).
%e A384178 The a(1) = 1 through a(14) = 8 strict partitions (A-E = 10-14):
%e A384178 1 2 3 4 5 6 7 8 9 A B C D E
%e A384178 21 32 321 43 431 54 532 65 543 76 653
%e A384178 421 521 432 541 542 651 643 743
%e A384178 621 721 632 732 652 761
%e A384178 4321 821 921 832 932
%e A384178 5321 6321 A21 B21
%e A384178 5431 5432
%e A384178 7321 8321
%t A384178 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Length/@Split[#,#1==#2+1&]&]],{n,0,30}]
%Y A384178 For subsets instead of strict partitions we have A384175, complement A384176.
%Y A384178 For anti-runs instead of runs we have A384880.
%Y A384178 This is the strict version of A384884.
%Y A384178 For equal instead of distinct lengths we have A384886.
%Y A384178 A000041 counts integer partitions, strict A000009.
%Y A384178 A047993 counts partitions with max part = length.
%Y A384178 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A384178 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384178 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384178 Cf. A008284, A044813, A325324, A325325, A329739, A351202.
%K A384178 nonn
%O A384178 0,4
%A A384178 _Gus Wiseman_, Jun 12 2025