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 A374766 #5 Aug 05 2024 08:44:26
%S A374766 1,0,1,0,0,2,0,0,1,3,0,0,0,3,5,0,0,0,1,8,7,0,0,0,1,3,17,11,0,0,0,0,4,
%T A374766 10,35,15,0,0,0,0,1,12,28,65,22,0,0,0,0,1,6,31,70,118,30,0,0,0,0,1,3,
%U A374766 22,78,163,203,42,0,0,0,0,0,4,13,69,186,354,342,56
%N A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.
%C A374766 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%C A374766 Are the column-sums finite?
%H A374766 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374766 Triangle begins:
%e A374766 1
%e A374766 0 1
%e A374766 0 0 2
%e A374766 0 0 1 3
%e A374766 0 0 0 3 5
%e A374766 0 0 0 1 8 7
%e A374766 0 0 0 1 3 17 11
%e A374766 0 0 0 0 4 10 35 15
%e A374766 0 0 0 0 1 12 28 65 22
%e A374766 0 0 0 0 1 6 31 70 118 30
%e A374766 0 0 0 0 1 3 22 78 163 203 42
%e A374766 0 0 0 0 0 4 13 69 186 354 342 56
%e A374766 Row n = 6 counts the following compositions:
%e A374766 . . . (321) (42) (51) (6)
%e A374766 (132) (411) (15)
%e A374766 (2121) (141) (24)
%e A374766 (312) (114)
%e A374766 (231) (33)
%e A374766 (213) (123)
%e A374766 (3111) (1113)
%e A374766 (1311) (222)
%e A374766 (1131) (1122)
%e A374766 (2211) (11112)
%e A374766 (2112) (111111)
%e A374766 (1221)
%e A374766 (1212)
%e A374766 (21111)
%e A374766 (12111)
%e A374766 (11211)
%e A374766 (11121)
%t A374766 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,Greater]]==k&]], {n,0,15},{k,0,n}]
%Y A374766 Column n = k is A000041.
%Y A374766 Row-sums are A011782.
%Y A374766 For length instead of sum we have A333213.
%Y A374766 The corresponding rank statistic is A374758, row-sums of A374757.
%Y A374766 For identical leaders we have A374760, ranks A374759.
%Y A374766 For distinct leaders we have A374761, ranks A374767.
%Y A374766 Other types of runs (instead of strictly decreasing):
%Y A374766 - For leaders of identical runs we have A373949.
%Y A374766 - For leaders of anti-runs we have A374521.
%Y A374766 - For leaders of weakly increasing runs we have A374637.
%Y A374766 - For leaders of strictly increasing runs we have A374700.
%Y A374766 - For leaders of weakly decreasing runs we have A374748.
%Y A374766 A003242 counts anti-run compositions.
%Y A374766 A238130, A238279, A333755 count compositions by number of runs.
%Y A374766 A335456 counts patterns matched by compositions.
%Y A374766 Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.
%K A374766 nonn,tabl
%O A374766 0,6
%A A374766 _Gus Wiseman_, Aug 02 2024