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A374700 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of strictly increasing runs sum to k.

%I A374700 #5 Jul 28 2024 10:39:01
%S A374700 1,0,1,0,0,2,0,1,0,3,0,1,2,0,5,0,1,3,5,0,7,0,2,4,6,9,0,11,0,2,7,10,13,
%T A374700 17,0,15,0,3,8,20,23,24,28,0,22,0,3,14,26,47,47,42,47,0,30,0,5,17,45,
%U A374700 66,101,92,71,73,0,42,0,5,27,61,124,154,201,166,116,114,0,56
%N A374700 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of strictly increasing runs sum to k.
%C A374700 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%H A374700 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374700 Triangle begins:
%e A374700    1
%e A374700    0   1
%e A374700    0   0   2
%e A374700    0   1   0   3
%e A374700    0   1   2   0   5
%e A374700    0   1   3   5   0   7
%e A374700    0   2   4   6   9   0  11
%e A374700    0   2   7  10  13  17   0  15
%e A374700    0   3   8  20  23  24  28   0  22
%e A374700    0   3  14  26  47  47  42  47   0  30
%e A374700    0   5  17  45  66 101  92  71  73   0  42
%e A374700    0   5  27  61 124 154 201 166 116 114   0  56
%e A374700    0   7  33 101 181 300 327 379 291 182 170   0  77
%e A374700    0   8  48 138 307 467 668 656 680 488 282 253   0 101
%e A374700 Row n = 6 counts the following compositions:
%e A374700   .  (15)   (24)    (231)   (312)    .  (6)
%e A374700      (123)  (141)   (213)   (2121)      (51)
%e A374700             (114)   (132)   (2112)      (42)
%e A374700             (1212)  (1311)  (1221)      (411)
%e A374700                     (1131)  (1122)      (33)
%e A374700                     (1113)  (12111)     (321)
%e A374700                             (11211)     (3111)
%e A374700                             (11121)     (222)
%e A374700                             (11112)     (2211)
%e A374700                                         (21111)
%e A374700                                         (111111)
%t A374700 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,Less]]==k&]],{n,0,15},{k,0,n}]
%Y A374700 Column n = k is A000041.
%Y A374700 Column k = 1 is A096765.
%Y A374700 Column k = 2 is A374705.
%Y A374700 Row-sums are A011782.
%Y A374700 For length instead of sum we have A333213.
%Y A374700 Leaders of strictly increasing runs in standard compositions are A374683.
%Y A374700 The corresponding rank statistic is A374684.
%Y A374700 Other types of runs (instead of strictly increasing):
%Y A374700 - For leaders of constant runs we have A373949.
%Y A374700 - For leaders of anti-runs we have A374521.
%Y A374700 - For leaders of weakly increasing runs we have A374637.
%Y A374700 - For leaders of weakly decreasing runs we have A374748.
%Y A374700 - For leaders of strictly decreasing runs we have A374766.
%Y A374700 A003242 counts anti-run compositions.
%Y A374700 A238130, A238279, A333755 count compositions by number of runs.
%Y A374700 A274174 counts contiguous compositions, ranks A374249.
%Y A374700 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374700 Cf. A106356, A124766, A238343, A261982, A374251, A374702, A374703.
%K A374700 nonn,tabl
%O A374700 0,6
%A A374700 _Gus Wiseman_, Jul 27 2024