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A363953
Number of integer partitions of n with high mode k.
8
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.
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.
Sequence in context: A361167 A319081 A336931 * A182662 A308778 A372472
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 07 2023
STATUS
approved