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 A383092 #12 Apr 26 2025 11:27:54
%S A383092 1,1,2,2,4,5,7,10,13,16,22,28,34,46,58,69,90,114,141,178,216,271,338,
%T A383092 418,506,630,769,941,1140,1399,1675,2051,2454,2975,3561,4289,5094,
%U A383092 6137,7274,8692,10269,12249,14414,17128,20110,23767,27872,32849,38346,45094,52552,61533
%N A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.
%F A383092 a(n) = A382915(n) + A383094(n).
%e A383092 The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
%e A383092 The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
%e A383092 The a(1) = 1 through a(7) = 10 partitions:
%e A383092 (1) (2) (3) (4) (5) (6) (7)
%e A383092 (11) (111) (22) (221) (33) (322)
%e A383092 (211) (311) (222) (331)
%e A383092 (1111) (2111) (411) (511)
%e A383092 (11111) (3111) (2221)
%e A383092 (21111) (4111)
%e A383092 (111111) (22111)
%e A383092 (31111)
%e A383092 (211111)
%e A383092 (1111111)
%t A383092 Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]
%Y A383092 For no choices we have A382915, ranks A382879.
%Y A383092 For at least one choice we have A383013, for run-sums A383098, ranks A383110.
%Y A383092 The complement is A383090, ranks A383089.
%Y A383092 Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.
%Y A383092 For a unique choice we have A383094, ranks A383112.
%Y A383092 For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.
%Y A383092 The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.
%Y A383092 Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
%Y A383092 A000041 counts integer partitions, strict A000009.
%Y A383092 A008284 counts partitions by length, strict A008289.
%Y A383092 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A383092 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383092 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A383092 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A383092 Cf. A006171, A047993, A362608, A381871, A382076, A382771, A383111.
%K A383092 nonn
%O A383092 0,3
%A A383092 _Gus Wiseman_, Apr 19 2025
%E A383092 More terms from _Bert Dobbelaere_, Apr 26 2025