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 A383094 #9 Apr 26 2025 08:06:17
%S A383094 1,1,2,2,4,4,5,6,9,7,11,10,13,12,17,14,21,16,21,18,27,22,29,22,34,25,
%T A383094 35,28,41,28,43,30,48,38,47,38,55,36,53,46,64,40,67,42,69,54,65,46,84,
%U A383094 51,75,62,83,52,86,62,94,70,83,58,111,60,89,80,106,74,115,66,111
%N A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.
%e A383094 The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
%e A383094 The a(1) = 1 through a(8) = 9 partitions:
%e A383094 (1) (2) (3) (4) (5) (6) (7) (8)
%e A383094 (11) (111) (22) (221) (33) (322) (44)
%e A383094 (211) (311) (222) (331) (332)
%e A383094 (1111) (11111) (411) (511) (422)
%e A383094 (111111) (22111) (611)
%e A383094 (1111111) (2222)
%e A383094 (22211)
%e A383094 (221111)
%e A383094 (11111111)
%t A383094 Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]
%Y A383094 The complement is ranked by A382879 \/ A383089.
%Y A383094 For no choices we have A382915, ranks A382879.
%Y A383094 For at least one choice we have A383013, for run-sums A383098, ranks A383110.
%Y A383094 For more than one choice we have A383090, ranks A383089.
%Y A383094 For at most one choice we have A383092, ranks A383091.
%Y A383094 For run-sums instead of lengths we have A383095, ranks A383099.
%Y A383094 Partitions of this type are ranked by A383112 = positions of 1 in A382857.
%Y A383094 Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
%Y A383094 A000041 counts integer partitions, strict A000009.
%Y A383094 A008284 counts partitions by length, strict A008289.
%Y A383094 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A383094 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383094 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A383094 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A383094 Cf. A047993, A362608, A381871, A382076, A382771, A383096, A383097, A383111.
%K A383094 nonn
%O A383094 0,3
%A A383094 _Gus Wiseman_, Apr 20 2025
%E A383094 More terms from _Bert Dobbelaere_, Apr 26 2025