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 A363263 #5 Jun 07 2023 08:32:00
%S A363263 0,1,1,1,2,3,2,4,4,5,7,10,8,13,13,15,19,25,24,35,35,43,50,61,59,79,83,
%T A363263 98,111,137,137,176,187,219,240,284,298,360,385,444,485,568,600,706,
%U A363263 763,867,951,1088,1168,1345,1453,1641,1792,2023,2179,2467,2673,2988
%N A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.
%C A363263 We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.
%e A363263 The a(1) = 1 through a(10) = 7 partitions:
%e A363263 1 11 111 211 221 21111 2221 22211 22221 33211
%e A363263 1111 2111 111111 22111 221111 32211 222211
%e A363263 11111 211111 2111111 2211111 322111
%e A363263 1111111 11111111 21111111 2221111
%e A363263 111111111 22111111
%e A363263 211111111
%e A363263 1111111111
%e A363263 The a(9) = 5 through a(12) = 8 partitions:
%e A363263 (22221) (33211) (33221) (2222211)
%e A363263 (32211) (222211) (222221) (3222111)
%e A363263 (2211111) (322111) (322211) (3321111)
%e A363263 (21111111) (2221111) (332111) (32211111)
%e A363263 (111111111) (22111111) (2222111) (222111111)
%e A363263 (211111111) (3221111) (2211111111)
%e A363263 (1111111111) (22211111) (21111111111)
%e A363263 (221111111) (111111111111)
%e A363263 (2111111111)
%e A363263 (11111111111)
%t A363263 comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A363263 Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]
%Y A363263 For parts instead of multiplicities we have A087897, complement A000009.
%Y A363263 For multisets instead of partitions we have A105039, complement A363224.
%Y A363263 The complement is counted by A363264.
%Y A363263 For mode we have A363484, complement A363485.
%Y A363263 A000041 counts integer partitions, A000009 covering an initial interval.
%Y A363263 A097979 counts normal multisets with a unique mode, complement A363262.
%Y A363263 A362607 counts partitions with multiple modes, co-modes A362609.
%Y A363263 A362608 counts partitions with a unique mode, co-mode A362610.
%Y A363263 A362614 counts partitions by number of modes, co-modes A362615.
%Y A363263 Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.
%K A363263 nonn
%O A363263 0,5
%A A363263 _Gus Wiseman_, Jun 06 2023