cp's OEIS Frontend

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

%I A364159 #11 Oct 18 2023 04:46:55
%S A364159 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 A364159 292,337,430,510,633,748,933,1083,1348,1579,1925,2262,2761,3197,3893,
%U A364159 4544,5458,6354,7634,8835,10577,12261,14546,16864,19990,23043,27226,31428
%N A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.
%C A364159 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 A364159 The a(1) = 1 through a(8) = 7 partitions:
%e A364159   (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
%e A364159              (111)  (1111)  (221)    (321)     (331)      (431)
%e A364159                             (11111)  (2211)    (421)      (521)
%e A364159                                      (111111)  (2221)     (3221)
%e A364159                                                (1111111)  (3311)
%e A364159                                                           (22211)
%e A364159                                                           (11111111)
%t A364159 Table[Length[Select[IntegerPartitions[n-1],Count[#,1]<Min@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]
%Y A364159 For mode instead of co-mode we have A241131, ranks A360015.
%Y A364159 The case with only one 1 is A364062, ranks A364061.
%Y A364159 Counts partitions ranked by A364158.
%Y A364159 Counts positions of 1's in A364191, high A364192.
%Y A364159 A362611 counts modes in prime factorization, triangle A362614.
%Y A364159 A362613 counts co-modes in prime factorization, triangle A362615.
%Y A364159 Ranking and counting partitions:
%Y A364159 - A356862 = unique mode, counted by A362608
%Y A364159 - A359178 = unique co-mode, counted by A362610
%Y A364159 - A362605 = multiple modes, counted by A362607
%Y A364159 - A362606 = multiple co-modes, counted by A362609
%Y A364159 Cf. A027336, A124943, A237984, A327472, A363486, A363487.
%K A364159 nonn
%O A364159 0,4
%A A364159 _Gus Wiseman_, Jul 16 2023