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Number of integer partitions of n where the least part is the unique mode.
2

%I #7 Jul 17 2023 17:59:34

%S 0,1,2,2,4,4,7,9,13,17,24,32,43,58,75,97,130,167,212,274,346,438,556,

%T 695,865,1082,1342,1655,2041,2511,3067,3756,4568,5548,6728,8130,9799,

%U 11810,14170,16980,20305,24251,28876,34366,40781,48342,57206,67597,79703

%N Number of integer partitions of n where the least part is the unique mode.

%C A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

%e The a(1) = 1 through a(8) = 13 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (311) (33) (322) (44)

%e (211) (2111) (222) (511) (422)

%e (1111) (11111) (411) (3211) (611)

%e (3111) (4111) (2222)

%e (21111) (22111) (4211)

%e (111111) (31111) (5111)

%e (211111) (32111)

%e (1111111) (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%t Table[If[n==0,0,Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]],{n,0,30}]

%Y For greatest part and multiple modes we have A171979.

%Y Allowing multiple modes gives A240303.

%Y For greatest instead of least part we have A362612, ranks A362616.

%Y For mean instead of least part we have A363723.

%Y These partitions have ranks A364160.

%Y A000041 counts integer partitions.

%Y A362611 counts modes in prime factorization, A362613 co-modes.

%Y A362614 counts partitions by number of modes, co-modes A362615.

%Y A363486 gives least mode in prime indices, A363487 greatest.

%Y A363952 counts partitions by low mode, A363953 high.

%Y Ranking and counting partitions:

%Y - A356862 = unique mode, counted by A362608

%Y - A359178 = unique co-mode, counted by A362610

%Y - A362605 = multiple modes, counted by A362607

%Y - A362606 = multiple co-modes, counted by A362609

%Y Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 16 2023