A363952 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

A363952

Number of integer partitions of n with low mode k.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 4, 2, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 9, 3, 2, 0, 0, 0, 1, 0, 13, 5, 2, 1, 0, 0, 0, 1, 0, 18, 6, 3, 2, 0, 0, 0, 0, 1, 0, 26, 9, 3, 2, 1, 0, 0, 0, 0, 1, 0, 32, 13, 5, 3, 2, 0, 0, 0, 0, 0, 1, 0, 47, 16, 7, 3, 2, 1, 0, 0, 0, 0, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

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}.

Extending the terminology of A124943, the "low mode" of a multiset is the least mode.

LINKS

Table of n, a(n) for n=0..90.

EXAMPLE

Triangle begins:

0 1

0 1 1

0 2 0 1

0 3 1 0 1

0 4 2 0 0 1

0 7 2 1 0 0 1

0 9 3 2 0 0 0 1

0 13 5 2 1 0 0 0 1

0 18 6 3 2 0 0 0 0 1

0 26 9 3 2 1 0 0 0 0 1

0 32 13 5 3 2 0 0 0 0 0 1

0 47 16 7 3 2 1 0 0 0 0 0 1

0 60 21 10 4 3 2 0 0 0 0 0 0 1

0 79 30 13 6 3 2 1 0 0 0 0 0 0 1

0 104 38 17 7 4 3 2 0 0 0 0 0 0 0 1

Row n = 8 counts the following partitions:

. (71) (62) (53) (44) . . . (8)

(611) (422) (332)

(521) (3221)

(5111) (2222)

(431) (22211)

(4211)

(41111)

(3311)

(32111)

(311111)

(221111)

(2111111)

(11111111)

MATHEMATICA

modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];

Table[Length[Select[IntegerPartitions[n], If[Length[#]==0, 0, First[modes[#]]]==k&]], {n, 0, 15}, {k, 0, n}]

CROSSREFS

Row sums are A000041.

For median: A124943 (high A124944), rank statistic A363941 (high A363942).

Column k = 1 is A241131 (partitions w/ low mode 1), ranks A360015, A360013.

The rank statistic for this triangle is A363486.

For mean: A363945 (high A363946), rank statistic A363943 (high A363944).

The high version is A363953.

A008284 counts partitions by length, A058398 by mean.

A362612 counts partitions (max part) = (unique mode), ranks A362616.

A362614 counts partitions by number of modes, rank statistic A362611.

A362615 counts partitions by number of co-modes, rank statistic A362613.

Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.

Cf. A002865, A025065, A026905, A067538, A237984, A363723, A363724, A363731.

Sequence in context: A359358 A318371 A363157 * A245151 A243978 A356898

Adjacent sequences: A363949 A363950 A363951 * A363953 A363954 A363955

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Jul 07 2023

STATUS

approved