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.

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, 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, 66, 101, 92, 71, 73, 0, 42, 0, 5, 27, 61, 124, 154, 201, 166, 116, 114, 0, 56

Offset

0,6

Comments

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.

Links

Table of n, a(n) for n=0..77.

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

Example

Triangle begins:
   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  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  66 101  92  71  73   0  42
   0   5  27  61 124 154 201 166 116 114   0  56
   0   7  33 101 181 300 327 379 291 182 170   0  77
   0   8  48 138 307 467 668 656 680 488 282 253   0 101
Row n = 6 counts the following compositions:
  .  (15)   (24)    (231)   (312)    .  (6)
     (123)  (141)   (213)   (2121)      (51)
            (114)   (132)   (2112)      (42)
            (1212)  (1311)  (1221)      (411)
                    (1131)  (1122)      (33)
                    (1113)  (12111)     (321)
                            (11211)     (3111)
                            (11121)     (222)
                            (11112)     (2211)
                                        (21111)
                                        (111111)

Mathematica

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

Crossrefs

Column n = k is A000041.

Column k = 1 is A096765.

Column k = 2 is A374705.

Row-sums are A011782.

For length instead of sum we have A333213.

Leaders of strictly increasing runs in standard compositions are A374683.

The corresponding rank statistic is A374684.

Other types of runs (instead of strictly increasing):

- For leaders of constant runs we have A373949.

- For leaders of anti-runs we have A374521.

- For leaders of weakly increasing runs we have A374637.

- 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.

A335548 counts non-contiguous compositions, ranks A374253.

Cf. A106356, A124766, A238343, A261982, A374251, A374702, A374703.

Sequence in context: A162515 A175267 A108045 * A298972 A143728 A127368

Adjacent sequences: A374697 A374698 A374699 * A374701 A374704 A374706

Keyword

nonn,tabl,new

Author

Gus Wiseman, Jul 27 2024

Status

approved