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Number of integer partitions of n - 1 containing fewer 1's than any other part.
6

%I #11 Oct 18 2023 04:46:55

%S 0,1,1,2,2,3,4,5,7,9,11,15,20,23,32,40,50,61,82,95,126,149,188,228,

%T 292,337,430,510,633,748,933,1083,1348,1579,1925,2262,2761,3197,3893,

%U 4544,5458,6354,7634,8835,10577,12261,14546,16864,19990,23043,27226,31428

%N Number of integer partitions of n - 1 containing fewer 1's than any other part.

%C Also integer partitions of n with least co-mode 1. Here, we define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.

%e The a(1) = 1 through a(8) = 7 partitions:

%e (1) (11) (21) (31) (41) (51) (61) (71)

%e (111) (1111) (221) (321) (331) (431)

%e (11111) (2211) (421) (521)

%e (111111) (2221) (3221)

%e (1111111) (3311)

%e (22211)

%e (11111111)

%t Table[Length[Select[IntegerPartitions[n-1],Count[#,1]<Min@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

%Y For mode instead of co-mode we have A241131, ranks A360015.

%Y The case with only one 1 is A364062, ranks A364061.

%Y Counts partitions ranked by A364158.

%Y Counts positions of 1's in A364191, high A364192.

%Y A362611 counts modes in prime factorization, triangle A362614.

%Y A362613 counts co-modes in prime factorization, triangle A362615.

%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. A027336, A124943, A237984, A327472, A363486, A363487.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 16 2023