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A363952
Number of integer partitions of n with low mode k.
9
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
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.
EXAMPLE
Triangle begins:
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
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.
Sequence in context: A359358 A318371 A363157 * A245151 A243978 A356898
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jul 07 2023
STATUS
approved