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

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 20, 23, 32, 40, 50, 61, 82, 95, 126, 149, 188, 228, 292, 337, 430, 510, 633, 748, 933, 1083, 1348, 1579, 1925, 2262, 2761, 3197, 3893, 4544, 5458, 6354, 7634, 8835, 10577, 12261, 14546, 16864, 19990, 23043, 27226, 31428

Offset

0,4

Comments

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}.

Links

Table of n, a(n) for n=0..51.

Example

The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (221)    (321)     (331)      (431)
                            (11111)  (2211)    (421)      (521)
                                     (111111)  (2221)     (3221)
                                               (1111111)  (3311)
                                                          (22211)
                                                          (11111111)

Mathematica

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

Crossrefs

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

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

Counts partitions ranked by A364158.

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

A362611 counts modes in prime factorization, triangle A362614.

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

Ranking and counting partitions:

- A356862 = unique mode, counted by A362608

- A359178 = unique co-mode, counted by A362610

- A362605 = multiple modes, counted by A362607

- A362606 = multiple co-modes, counted by A362609

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

Sequence in context: A123946 A002569 A129528 * A280200 A052336 A318053

Adjacent sequences: A364156 A364157 A364158 * A364160 A364161 A364162

Keyword

nonn

Author

Gus Wiseman, Jul 16 2023

Status

approved