login
A362614
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.
54
1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25
OFFSET
0,5
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.
LINKS
FORMULA
Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024
EXAMPLE
Triangle begins:
1
0 1
0 2
0 2 1
0 4 1
0 5 2
0 7 3 1
0 11 3 1
0 16 4 2
0 21 6 3
0 29 8 4 1
0 43 7 5 1
0 54 13 8 2
0 78 12 8 3
0 102 17 11 5
0 131 26 12 6 1
0 175 29 17 9 1
Row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(332) (71)
(422) (3311)
(611)
(2222)
(3221)
(4211)
(5111)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
msi[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[msi[#]]==k&]], {n, 0, 15}, {k, 0, Floor[(Sqrt[1+8n]-1)/2]}]
CROSSREFS
Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Sequence in context: A080844 A321428 A362615 * A336167 A076626 A182886
KEYWORD
nonn,look,tabf
AUTHOR
Gus Wiseman, May 04 2023
STATUS
approved