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.

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, 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, 107, 60, 60, 48, 9, 14, 4

Offset: 0

Author

Gus Wiseman, Aug 02 2024

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
   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   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  88 107  60  60  48   9  14   4
   0  78 144 166 179 176 101  83  68  10  17   2
   0 136 258 327 339 311 299 139 122  81  12  18   6
   0 242 457 602 704 591 544 447 198 165 109  12  23   2
Row n = 6 counts the following compositions:
  .  (15)    (24)    (321)    (42)     (51)     (6)
     (141)   (114)   (312)    (1122)   (411)    (33)
     (132)   (231)   (1113)   (11112)  (3111)   (222)
     (123)   (213)   (2112)            (2211)   (111111)
     (1212)  (1311)  (1221)            (21111)
             (1131)  (12111)
             (2121)  (11211)
                     (11121)
		

Links

• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Column n = k is A000005, except a(0) = 1.

Row-sums are A011782.

Column k = 1 is A096569.

For length instead of sum we have A106356.

The corresponding rank statistic is A374516, row-sums of A374515.

For identical leaders we have A374517, ranks A374519.

For distinct leaders we have A374518, ranks A374638.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A373949.

- For leaders of weakly increasing runs we have A374637.

- For leaders of strictly increasing runs we have A374700.

- For leaders of weakly decreasing runs we have A374748.

- For leaders of strictly decreasing runs we have A374766.

A003242 counts anti-run compositions.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

Cf. A124766, A238343, A261982, A333213, A374251, A374687, A374761.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,UnsameQ]]==k&]],{n,0,15},{k,0,n}]