A363952 - OEIS

%I #7 Jul 07 2023 23:09:10

%S 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,

%T 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,

%U 32,13,5,3,2,0,0,0,0,0,1,0,47,16,7,3,2,1,0,0,0,0,0,1

%N Number of integer partitions of n with low mode k.

%C 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}.

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

%e Triangle begins:

%e    1

%e    0   1

%e    0   1   1

%e    0   2   0   1

%e    0   3   1   0   1

%e    0   4   2   0   0   1

%e    0   7   2   1   0   0   1

%e    0   9   3   2   0   0   0   1

%e    0  13   5   2   1   0   0   0   1

%e    0  18   6   3   2   0   0   0   0   1

%e    0  26   9   3   2   1   0   0   0   0   1

%e    0  32  13   5   3   2   0   0   0   0   0   1

%e    0  47  16   7   3   2   1   0   0   0   0   0   1

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

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

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

%e Row n = 8 counts the following partitions:

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

%e      (611)       (422)    (332)

%e      (521)       (3221)

%e      (5111)      (2222)

%e      (431)       (22211)

%e      (4211)

%e      (41111)

%e      (3311)

%e      (32111)

%e      (311111)

%e      (221111)

%e      (2111111)

%e      (11111111)

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

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

%Y Row sums are A000041.

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

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

%Y The rank statistic for this triangle is A363486.

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

%Y The high version is A363953.

%Y A008284 counts partitions by length, A058398 by mean.

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

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

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

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

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

%K nonn,tabl

%O 0,8

%A _Gus Wiseman_, Jul 07 2023