This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363953 #6 Jul 07 2023 23:08:50
%S A363953 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,
%T A363953 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,
%U A363953 26,10,5,2,3,3,2,2,1,1,1
%N A363953 Number of integer partitions of n with high mode k.
%C A363953 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 A363953 Extending the terminology of A124944, the "high mode" in a multiset is the greatest mode.
%e A363953 Triangle begins:
%e A363953 1
%e A363953 0 1
%e A363953 0 1 1
%e A363953 0 1 1 1
%e A363953 0 2 1 1 1
%e A363953 0 3 1 1 1 1
%e A363953 0 4 2 2 1 1 1
%e A363953 0 7 2 1 2 1 1 1
%e A363953 0 9 4 2 2 2 1 1 1
%e A363953 0 13 6 2 2 2 2 1 1 1
%e A363953 0 18 7 4 3 3 2 2 1 1 1
%e A363953 0 26 10 5 2 3 3 2 2 1 1 1
%e A363953 0 32 15 8 4 4 4 3 2 2 1 1 1
%e A363953 0 47 19 9 5 3 4 4 3 2 2 1 1 1
%e A363953 0 60 26 13 7 5 5 5 4 3 2 2 1 1 1
%e A363953 0 79 34 18 10 6 5 5 5 4 3 2 2 1 1 1
%e A363953 Row n = 9 counts the following partitions:
%e A363953 . (711) (522) (333) (441) (54) (63) (72) (81) (9)
%e A363953 (6111) (4221) (3321) (432) (531) (621)
%e A363953 (5211) (3222)
%e A363953 (51111) (32211)
%e A363953 (4311) (22221)
%e A363953 (42111) (222111)
%e A363953 (411111)
%e A363953 (33111)
%e A363953 (321111)
%e A363953 (3111111)
%e A363953 (2211111)
%e A363953 (21111111)
%e A363953 (111111111)
%t A363953 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363953 Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,Last[modes[#]]]==k&]],{n,0,15},{k,0,n}]
%Y A363953 Row sums are A000041.
%Y A363953 For median: A124944 (low A124943), rank statistic A363942 (low A363941).
%Y A363953 Column k = 1 is A241131 (partitions w/ high mode 1), ranks A360013, A360015.
%Y A363953 The rank statistic for this triangle is A363487, low A363486.
%Y A363953 For mean: A363946 (low A363945), rank statistic A363944 (low A363943).
%Y A363953 The low version is A363952.
%Y A363953 A008284 counts partitions by length, A058398 by mean.
%Y A363953 A362612 counts partitions (max part) = (unique mode), ranks A362616.
%Y A363953 A362614 counts partitions by number of modes, rank statistic A362611.
%Y A363953 A362615 counts partitions by number of co-modes, rank statistic A362613.
%Y A363953 Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
%Y A363953 Cf. A002865, A025065, A026905, A237984, A363723, A363724, A363731.
%K A363953 nonn
%O A363953 0,12
%A A363953 _Gus Wiseman_, Jul 07 2023