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 A363952 #7 Jul 07 2023 23:09:10
%S A363952 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 A363952 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 A363952 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 A363952 Number of integer partitions of n with low mode k.
%C A363952 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 A363952 Extending the terminology of A124943, the "low mode" of a multiset is the least mode.
%e A363952 Triangle begins:
%e A363952 1
%e A363952 0 1
%e A363952 0 1 1
%e A363952 0 2 0 1
%e A363952 0 3 1 0 1
%e A363952 0 4 2 0 0 1
%e A363952 0 7 2 1 0 0 1
%e A363952 0 9 3 2 0 0 0 1
%e A363952 0 13 5 2 1 0 0 0 1
%e A363952 0 18 6 3 2 0 0 0 0 1
%e A363952 0 26 9 3 2 1 0 0 0 0 1
%e A363952 0 32 13 5 3 2 0 0 0 0 0 1
%e A363952 0 47 16 7 3 2 1 0 0 0 0 0 1
%e A363952 0 60 21 10 4 3 2 0 0 0 0 0 0 1
%e A363952 0 79 30 13 6 3 2 1 0 0 0 0 0 0 1
%e A363952 0 104 38 17 7 4 3 2 0 0 0 0 0 0 0 1
%e A363952 Row n = 8 counts the following partitions:
%e A363952 . (71) (62) (53) (44) . . . (8)
%e A363952 (611) (422) (332)
%e A363952 (521) (3221)
%e A363952 (5111) (2222)
%e A363952 (431) (22211)
%e A363952 (4211)
%e A363952 (41111)
%e A363952 (3311)
%e A363952 (32111)
%e A363952 (311111)
%e A363952 (221111)
%e A363952 (2111111)
%e A363952 (11111111)
%t A363952 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363952 Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,First[modes[#]]]==k&]],{n,0,15},{k,0,n}]
%Y A363952 Row sums are A000041.
%Y A363952 For median: A124943 (high A124944), rank statistic A363941 (high A363942).
%Y A363952 Column k = 1 is A241131 (partitions w/ low mode 1), ranks A360015, A360013.
%Y A363952 The rank statistic for this triangle is A363486.
%Y A363952 For mean: A363945 (high A363946), rank statistic A363943 (high A363944).
%Y A363952 The high version is A363953.
%Y A363952 A008284 counts partitions by length, A058398 by mean.
%Y A363952 A362612 counts partitions (max part) = (unique mode), ranks A362616.
%Y A363952 A362614 counts partitions by number of modes, rank statistic A362611.
%Y A363952 A362615 counts partitions by number of co-modes, rank statistic A362613.
%Y A363952 Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
%Y A363952 Cf. A002865, A025065, A026905, A067538, A237984, A363723, A363724, A363731.
%K A363952 nonn,tabl
%O A363952 0,8
%A A363952 _Gus Wiseman_, Jul 07 2023