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A374637 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly increasing runs sum to k.

%I A374637 #9 Jul 24 2024 09:21:08
%S A374637 1,0,1,0,1,1,0,2,0,2,0,3,2,1,2,0,5,4,3,1,3,0,7,10,7,3,1,4,0,11,19,14,
%T A374637 9,4,2,5,0,15,39,27,22,10,7,2,6,0,22,69,59,48,24,15,8,3,8,0,30,125,
%U A374637 117,104,56,38,19,10,3,10,0,42,211,241,215,132,80,49,25,12,5,12
%N A374637 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly increasing runs sum to k.
%C A374637 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%H A374637 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374637 Triangle begins:
%e A374637    1
%e A374637    0   1
%e A374637    0   1   1
%e A374637    0   2   0   2
%e A374637    0   3   2   1   2
%e A374637    0   5   4   3   1   3
%e A374637    0   7  10   7   3   1   4
%e A374637    0  11  19  14   9   4   2   5
%e A374637    0  15  39  27  22  10   7   2   6
%e A374637    0  22  69  59  48  24  15   8   3   8
%e A374637    0  30 125 117 104  56  38  19  10   3  10
%e A374637    0  42 211 241 215 132  80  49  25  12   5  12
%e A374637    0  56 354 473 445 296 186 109  61  31  17   5  15
%e A374637    0  77 571 917 896 665 409 258 139  78  41  20   7  18
%e A374637 Row n = 6 counts the following compositions:
%e A374637   .  (15)      (24)     (33)     (312)   (411)  (6)
%e A374637      (114)     (141)    (231)    (3111)         (51)
%e A374637      (123)     (1311)   (213)    (2121)         (42)
%e A374637      (1113)    (1131)   (132)                   (321)
%e A374637      (1122)    (222)    (2211)
%e A374637      (11112)   (1221)   (2112)
%e A374637      (111111)  (1212)   (21111)
%e A374637                (12111)
%e A374637                (11211)
%e A374637                (11121)
%t A374637 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,LessEqual]]==k&]],{n,0,15},{k,0,n}]
%Y A374637 Last column n = k is A000009.
%Y A374637 Second column k = 2 is A000041.
%Y A374637 Row-sums are A011782.
%Y A374637 For length instead of sum we have A238343.
%Y A374637 The corresponding rank statistic is A374630, row-sums of A374629.
%Y A374637 Types of runs (instead of weakly increasing):
%Y A374637 - For leaders of constant runs we have A373949.
%Y A374637 - For leaders of anti-runs we have A374521.
%Y A374637 - For leaders of strictly increasing runs we have A374700.
%Y A374637 - For leaders of weakly decreasing runs we have A374748.
%Y A374637 - For leaders of strictly decreasing runs we have A374766.
%Y A374637 Types of run-leaders:
%Y A374637 - For strictly decreasing leaders we appear to have A188920.
%Y A374637 - For weakly decreasing leaders we appear to have A189076.
%Y A374637 - For identical leaders we have A374631.
%Y A374637 - For distinct leaders we have A374632, ranks A374768.
%Y A374637 - For strictly increasing leaders we have A374634.
%Y A374637 - For weakly increasing leaders we have A374635.
%Y A374637 A003242 counts anti-run compositions.
%Y A374637 A238130, A238279, A333755 count compositions by number of runs.
%Y A374637 A274174 counts contiguous compositions, ranks A374249.
%Y A374637 A335456 counts patterns matched by compositions.
%Y A374637 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374637 Cf. A106356, A124766, A261982, A333213, A374251, A374518, A374687, A374761.
%K A374637 nonn,tabl
%O A374637 0,8
%A A374637 _Gus Wiseman_, Jul 23 2024