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 A383013 #17 Apr 27 2025 15:04:34
%S A383013 1,1,2,3,5,6,9,11,18,21,31,38,56,67,94,121,162,199,265,330,438,543,
%T A383013 693,859,1103,1353,1702,2097,2619,3194,3972,4821,5943,7206,8796,10632,
%U A383013 12938,15536,18794,22539,27133,32374,38827,46175,55134,65421,77751,91951,109011,128482
%N A383013 Number of integer partitions of n having a permutation with all equal run-lengths.
%C A383013 A partition of n counts towards a(n) if and only if #p + g >= 2*L where #p is the number of parts counted with multiplicity of the partition, g is the gcd of all the frequencies of every distinct part and L is the largest frequency of a part. - _David A. Corneth_, Apr 27 2025
%H A383013 David A. Corneth, <a href="/A383013/b383013.txt">Table of n, a(n) for n = 0..82</a>
%e A383013 The partition (2,2,1,1,1,1) has permutation (1,1,2,2,1,1) with equal run-lengths (2,2,2) so is counted under a(8).
%e A383013 The a(1) = 1 through a(8) = 18 partitions:
%e A383013 (1) (2) (3) (4) (5) (6) (7) (8)
%e A383013 (11) (21) (22) (32) (33) (43) (44)
%e A383013 (111) (31) (41) (42) (52) (53)
%e A383013 (211) (221) (51) (61) (62)
%e A383013 (1111) (311) (222) (322) (71)
%e A383013 (11111) (321) (331) (332)
%e A383013 (411) (421) (422)
%e A383013 (2211) (511) (431)
%e A383013 (111111) (3211) (521)
%e A383013 (22111) (611)
%e A383013 (1111111) (2222)
%e A383013 (3221)
%e A383013 (3311)
%e A383013 (4211)
%e A383013 (22211)
%e A383013 (32111)
%e A383013 (221111)
%e A383013 (11111111)
%t A383013 Table[Length[Select[IntegerPartitions[n],Select[Permutations[#], SameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]
%Y A383013 For distinct instead of equal run-lengths we have A239455, ranked by A351294.
%Y A383013 The complement for distinct run-lengths is A351293, ranked by A351295.
%Y A383013 The complement is counted by A382915, ranked by A382879, by signature A382914.
%Y A383013 A000041 counts integer partitions, strict A000009.
%Y A383013 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A383013 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383013 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A383013 A382857 counts permutations of prime indices with equal run-lengths, firsts A382878.
%Y A383013 Cf. A003242, A047966, A164707, A238279, A351201, A351290, A351291, A353837, A353851, A382858.
%K A383013 nonn
%O A383013 0,3
%A A383013 _Gus Wiseman_, Apr 12 2025
%E A383013 More terms from _Bert Dobbelaere_, Apr 26 2025