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A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

%I A374748 #10 Sep 16 2024 08:43:04
%S A374748 1,0,1,0,1,1,0,1,1,2,0,1,2,3,2,0,1,2,6,4,3,0,1,3,9,8,7,4,0,1,3,13,15,
%T A374748 16,11,5,0,1,4,17,24,32,28,16,6,0,1,4,23,36,58,58,44,24,8,0,1,5,28,52,
%U A374748 96,115,100,71,34,10,0,1,5,35,72,151,203,211,176,109,49,12
%N A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.
%C A374748 The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
%H A374748 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374748 Triangle begins:
%e A374748    1
%e A374748    0   1
%e A374748    0   1   1
%e A374748    0   1   1   2
%e A374748    0   1   2   3   2
%e A374748    0   1   2   6   4   3
%e A374748    0   1   3   9   8   7   4
%e A374748    0   1   3  13  15  16  11   5
%e A374748    0   1   4  17  24  32  28  16   6
%e A374748    0   1   4  23  36  58  58  44  24   8
%e A374748    0   1   5  28  52  96 115 100  71  34  10
%e A374748    0   1   5  35  72 151 203 211 176 109  49  12
%e A374748 Row n = 6 counts the following compositions:
%e A374748   .  (111111)  (222)    (33)     (42)    (51)    (6)
%e A374748                (2211)   (321)    (411)   (141)   (15)
%e A374748                (21111)  (3111)   (132)   (114)   (24)
%e A374748                         (1221)   (1311)  (312)   (123)
%e A374748                         (1122)   (1131)  (231)
%e A374748                         (12111)  (1113)  (213)
%e A374748                         (11211)  (2121)  (1212)
%e A374748                         (11121)  (2112)
%e A374748                         (11112)
%t A374748 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==k&]],{n,0,15},{k,0,n}]
%Y A374748 Column n = k is A000009.
%Y A374748 Column k = 2 is A004526.
%Y A374748 Row-sums are A011782.
%Y A374748 For length instead of sum we have A238343.
%Y A374748 The opposite rank statistic is A374630, row-sums of A374629.
%Y A374748 Column k = 3 is A374702.
%Y A374748 The center n = 2k is A374703.
%Y A374748 The corresponding rank statistic is A374741 row-sums of A374740.
%Y A374748 Types of runs (instead of weakly decreasing):
%Y A374748 - For leaders of constant runs we have A373949.
%Y A374748 - For leaders of anti-runs we have A374521.
%Y A374748 - For leaders of weakly increasing runs we have A374637.
%Y A374748 - For leaders of strictly increasing runs we have A374700.
%Y A374748 - For leaders of strictly decreasing runs we have A374766.
%Y A374748 Types of run-leaders:
%Y A374748 - For weakly increasing leaders we appear to have A188900.
%Y A374748 - For identical leaders we have A374742, ranks A374744.
%Y A374748 - For distinct leaders we have A374743, ranks A374701.
%Y A374748 - For strictly decreasing leaders we have A374746.
%Y A374748 - For weakly decreasing leaders we have A374747.
%Y A374748 A003242 counts anti-run compositions.
%Y A374748 A238130, A238279, A333755 count compositions by number of runs.
%Y A374748 A274174 counts contiguous compositions, ranks A374249.
%Y A374748 A335456 counts patterns matched by compositions.
%Y A374748 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374748 Cf. A000041, A106356, A124766, A261982, A333213, A374251, A374761.
%K A374748 nonn,tabl
%O A374748 0,10
%A A374748 _Gus Wiseman_, Jul 26 2024