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A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

%I A374521 #6 Aug 05 2024 08:44:40
%S A374521 1,0,1,0,0,2,0,1,1,2,0,2,1,2,3,0,2,5,3,4,2,0,5,7,8,3,5,4,0,9,12,11,17,
%T A374521 5,8,2,0,14,26,23,22,24,6,9,4,0,25,42,54,41,36,36,7,12,3,0,46,76,88,
%U A374521 107,60,60,48,9,14,4
%N A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.
%C A374521 The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%H A374521 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374521 Triangle begins:
%e A374521    1
%e A374521    0   1
%e A374521    0   0   2
%e A374521    0   1   1   2
%e A374521    0   2   1   2   3
%e A374521    0   2   5   3   4   2
%e A374521    0   5   7   8   3   5   4
%e A374521    0   9  12  11  17   5   8   2
%e A374521    0  14  26  23  22  24   6   9   4
%e A374521    0  25  42  54  41  36  36   7  12   3
%e A374521    0  46  76  88 107  60  60  48   9  14   4
%e A374521    0  78 144 166 179 176 101  83  68  10  17   2
%e A374521    0 136 258 327 339 311 299 139 122  81  12  18   6
%e A374521    0 242 457 602 704 591 544 447 198 165 109  12  23   2
%e A374521 Row n = 6 counts the following compositions:
%e A374521   .  (15)    (24)    (321)    (42)     (51)     (6)
%e A374521      (141)   (114)   (312)    (1122)   (411)    (33)
%e A374521      (132)   (231)   (1113)   (11112)  (3111)   (222)
%e A374521      (123)   (213)   (2112)            (2211)   (111111)
%e A374521      (1212)  (1311)  (1221)            (21111)
%e A374521              (1131)  (12111)
%e A374521              (2121)  (11211)
%e A374521                      (11121)
%t A374521 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,UnsameQ]]==k&]],{n,0,15},{k,0,n}]
%Y A374521 Column n = k is A000005, except a(0) = 1.
%Y A374521 Row-sums are A011782.
%Y A374521 Column k = 1 is A096569.
%Y A374521 For length instead of sum we have A106356.
%Y A374521 The corresponding rank statistic is A374516, row-sums of A374515.
%Y A374521 For identical leaders we have A374517, ranks A374519.
%Y A374521 For distinct leaders we have A374518, ranks A374638.
%Y A374521 Other types of runs (instead of anti-):
%Y A374521 - For leaders of identical runs we have A373949.
%Y A374521 - For leaders of weakly increasing runs we have A374637.
%Y A374521 - For leaders of strictly increasing runs we have A374700.
%Y A374521 - For leaders of weakly decreasing runs we have A374748.
%Y A374521 - For leaders of strictly decreasing runs we have A374766.
%Y A374521 A003242 counts anti-run compositions.
%Y A374521 A238130, A238279, A333755 count compositions by number of runs.
%Y A374521 A274174 counts contiguous compositions, ranks A374249.
%Y A374521 Cf. A124766, A238343, A261982, A333213, A374251, A374687, A374761.
%K A374521 nonn,tabl
%O A374521 0,6
%A A374521 _Gus Wiseman_, Aug 02 2024