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A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

%I A383111 #9 May 04 2025 06:35:13
%S A383111 0,0,0,0,1,3,3,8,9,13,17,26,27,43,51,61,78,103,115,153,174,213,255,
%T A383111 316,354,442,508,610,701,848,950,1153,1303,1539,1750,2075,2318,2738,
%U A383111 3081
%N A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.
%e A383111 The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
%e A383111 The a(4) = 1 through a(9) = 13 partitions:
%e A383111   (211)  (221)   (411)    (322)     (332)      (441)
%e A383111          (311)   (3111)   (331)     (422)      (522)
%e A383111          (2111)  (21111)  (511)     (611)      (711)
%e A383111                           (2221)    (5111)     (3222)
%e A383111                           (4111)    (22211)    (6111)
%e A383111                           (22111)   (41111)    (22221)
%e A383111                           (31111)   (221111)   (33111)
%e A383111                           (211111)  (311111)   (51111)
%e A383111                                     (2111111)  (222111)
%e A383111                                                (411111)
%e A383111                                                (2211111)
%e A383111                                                (3111111)
%e A383111                                                (21111111)
%t A383111 Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], UnsameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]
%Y A383111 For a unique choice we have A000005, ranks A000961.
%Y A383111 For at least one choice we have A239455, ranks A351294, conjugate A381432.
%Y A383111 For no choices we have A351293, ranks A351295, conjugate A381433.
%Y A383111 The complement is A351293 + A000005, ranks too dense.
%Y A383111 For equal instead of distinct run-lengths we have A383090, ranks A383089.
%Y A383111 These partitions are ranked by A383113 = positions of terms > 1 in A382771.
%Y A383111 Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883
%Y A383111 A000041 counts integer partitions, strict A000009.
%Y A383111 A008284 counts partitions by length, strict A008289.
%Y A383111 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383111 Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.
%K A383111 nonn,hard,more
%O A383111 0,6
%A A383111 _Gus Wiseman_, Apr 20 2025
%E A383111 a(21)-a(38) from _Jakub Buczak_, May 04 2025