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%I #11 Apr 26 2025 11:28:02
%S 0,0,0,0,0,1,2,4,4,9,11,18,21,34,41,55,69,98,120,160,189,249,309,396,
%T 472,605,734,913,1099,1371,1632,2021,2406,2937,3514,4251,5039,6101,
%U 7221,8646,10205,12209,14347,17086,20041,23713,27807,32803,38262,45043,52477,61471,71496
%N Number of integer partitions of n having no permutation with all equal run-lengths.
%e The partition y = (2,2,1,1,1) has permutations and run-lengths:
%e (2,2,1,1,1) (2,3)
%e (2,1,2,1,1) (1,1,1,2)
%e (2,1,1,2,1) (1,2,1,1)
%e (2,1,1,1,2) (1,3,1)
%e (1,2,2,1,1) (1,2,2)
%e (1,2,1,2,1) (1,1,1,1,1)
%e (1,2,1,1,2) (1,1,2,1)
%e (1,1,2,2,1) (2,2,1)
%e (1,1,2,1,2) (2,1,1,1)
%e (1,1,1,2,2) (3,2)
%e Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
%e The a(5) = 1 through a(10) = 11 partitions:
%e (2111) (3111) (2221) (5111) (3222) (3331)
%e (21111) (4111) (41111) (6111) (4222)
%e (31111) (311111) (22221) (7111)
%e (211111) (2111111) (51111) (61111)
%e (321111) (421111)
%e (411111) (511111)
%e (2211111) (3211111)
%e (3111111) (4111111)
%e (21111111) (22111111)
%e (31111111)
%e (211111111)
%t Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Length/@Split[#]&]=={}&]],{n,0,15}]
%Y The complement for distinct run-lengths is A239455, ranked by A351294.
%Y For distinct instead of equal run-lengths we have A351293, ranked by A351295.
%Y These partitions are ranked by A382879, by signature A382914.
%Y The complement is counted by A383013.
%Y A000041 counts integer partitions, strict A000009.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A304442 counts partitions with equal run-sums, ranks A353833.
%Y A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A382857 counts permutations of prime indices with equal run-lengths.
%Y Cf. A003242, A047966, A238279, A329739, A351201, A351290, A351596, A382773.
%K nonn
%O 0,7
%A _Gus Wiseman_, Apr 12 2025
%E More terms from _Bert Dobbelaere_, Apr 26 2025