A363953 Number of integer partitions of n with high mode k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 2, 2, 1, 1, 1, 0, 7, 2, 1, 2, 1, 1, 1, 0, 9, 4, 2, 2, 2, 1, 1, 1, 0, 13, 6, 2, 2, 2, 2, 1, 1, 1, 0, 18, 7, 4, 3, 3, 2, 2, 1, 1, 1, 0, 26, 10, 5, 2, 3, 3, 2, 2, 1, 1, 1

Offset

0,12

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 A124944, the "high mode" in a multiset is the greatest mode.

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  3  1  1  1  1
  0  4  2  2  1  1  1
  0  7  2  1  2  1  1  1
  0  9  4  2  2  2  1  1  1
  0 13  6  2  2  2  2  1  1  1
  0 18  7  4  3  3  2  2  1  1  1
  0 26 10  5  2  3  3  2  2  1  1  1
  0 32 15  8  4  4  4  3  2  2  1  1  1
  0 47 19  9  5  3  4  4  3  2  2  1  1  1
  0 60 26 13  7  5  5  5  4  3  2  2  1  1  1
  0 79 34 18 10  6  5  5  5  4  3  2  2  1  1  1
Row n = 9 counts the following partitions:
  .  (711)        (522)     (333)   (441)  (54)   (63)   (72)  (81)  (9)
     (6111)       (4221)    (3321)  (432)  (531)  (621)
     (5211)       (3222)
     (51111)      (32211)
     (4311)       (22221)
     (42111)      (222111)
     (411111)
     (33111)
     (321111)
     (3111111)
     (2211111)
     (21111111)
     (111111111)

Mathematica

modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], If[Length[#]==0, 0, Last[modes[#]]]==k&]], {n, 0, 15}, {k, 0, n}]

Crossrefs

Row sums are A000041.

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

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

The rank statistic for this triangle is A363487, low A363486.

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

The low version is A363952.

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, A237984, A363723, A363724, A363731.

Sequence in context: A361167 A319081 A336931 * A182662 A308778 A372472

Adjacent sequences: A363950 A363951 A363952 * A363954 A363955 A363956

Keyword

nonn

Author

Gus Wiseman, Jul 07 2023

Status

approved