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 A383090 #13 Apr 26 2025 11:27:59
%S A383090 0,0,0,1,1,2,4,5,9,14,20,28,43,55,77,107,141,183,244,312,411,521,664,
%T A383090 837,1069,1328,1667,2069,2578,3166,3929,4791,5895,7168,8749,10594,
%U A383090 12883,15500,18741,22493,27069,32334,38760,46133,55065,65367,77686,91905,108927,128431,151674
%N A383090 Number of integer partitions of n having more than one permutation with all equal run-lengths.
%F A383090 The complement is counted by A383094 + A382915, ranks A383112 \/ A382879.
%e A383090 The partition (3322221) has 3 permutations with all equal run-lengths: (2323212), (2321232), (2123232), so is counted under a(15).
%e A383090 The partition (3322111111) has 2 permutations with all equal run-lengths: (1133112211), (1122113311), so is counted under a(16).
%e A383090 The a(3) = 1 through a(9) = 14 partitions:
%e A383090 (21) (31) (32) (42) (43) (53) (54)
%e A383090 (41) (51) (52) (62) (63)
%e A383090 (321) (61) (71) (72)
%e A383090 (2211) (421) (431) (81)
%e A383090 (3211) (521) (432)
%e A383090 (3221) (531)
%e A383090 (3311) (621)
%e A383090 (4211) (3321)
%e A383090 (32111) (4221)
%e A383090 (4311)
%e A383090 (5211)
%e A383090 (32211)
%e A383090 (42111)
%e A383090 (222111)
%t A383090 Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]
%Y A383090 For no choices we have A382915, ranks A382879.
%Y A383090 For at least one choice we have A383013, for run-sums A383098, ranks A383110.
%Y A383090 Partitions of this type are ranked by A383089 = positions of terms > 1 in A382857.
%Y A383090 The complement is A383091, counted by A383092.
%Y A383090 For a unique choice we have A383094, ranks A383112.
%Y A383090 The complement for run-sums is A383095 + A383096, ranks A383099 \/ A383100.
%Y A383090 For run-sums we have A383097, ranked by A383015 = positions of terms > 1 in A382877.
%Y A383090 For distinct instead of equal run-lengths we have A383111, ranks A383113.
%Y A383090 Cf. A047966, A072774, A098859, A304442, A100471, A100881, A100882, A100883.
%Y A383090 A000041 counts integer partitions, strict A000009.
%Y A383090 A008284 counts partitions by length, strict A008289.
%Y A383090 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A383090 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383090 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A383090 Cf. A047993, A237984, A329739, A381541, A381871, A382076, A382771.
%K A383090 nonn
%O A383090 0,6
%A A383090 _Gus Wiseman_, Apr 19 2025
%E A383090 More terms from _Bert Dobbelaere_, Apr 26 2025