A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.

1, 1, 2, 3, 6, 9, 17, 29, 51, 91, 162, 291, 523, 948, 1712, 3112, 5656, 10297, 18763, 34217, 62442, 114006, 208239, 380465, 695342, 1271046, 2323818, 4249113, 7770389, 14210991, 25991853, 47541734, 86962675, 159077005, 291001483, 532345978, 973871397

Offset: 0

Author

Gus Wiseman, Jul 27 2024

Keywords

• nonn

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are identical. For maxima instead of minima we have A374760. For all partitions (not just strict) we have A374704, for maxima A358905.

Examples

			The composition (2,3,2,2,3,4) has strictly increasing runs ((2,3),(2),(2,3,4)), with leaders (2,2,2), so is counted under a(16).
The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (131)    (114)
                        (1111)  (1112)   (123)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1131)
                                         (1212)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Ranked by A374685.

Types of runs (instead of strictly increasing):

- For leaders of identical runs we have A000005 for n > 0, ranks A272919.

- For leaders of anti-runs we have A374517, ranks A374519.

- For leaders of weakly increasing runs we have A374631, ranks A374633.

- For leaders of weakly decreasing runs we have A374742, ranks A374744.

- For leaders of strictly decreasing runs we have A374760, ranks A374759.

Types of run-leaders (instead of identical):

- For distinct leaders we have A374687, ranks A374698.

- For strictly increasing leaders we have A374688.

- For strictly decreasing leaders we have A374689.

- For weakly increasing leaders we have A374690.

- For weakly decreasing leaders we have A374697.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

A335456 counts patterns matched by compositions.

A335548 counts non-contiguous compositions, ranks A374253.

A373949 counts compositions by run-compressed sum, opposite A373951.

A374683 lists leaders of strictly increasing runs of standard compositions.

A374700 counts compositions by sum of leaders of strictly increasing runs.

Cf. A000009, A106356, A188920, A189076, A238343, A304969, A333213, A374632, A374634, A374635, A374640.

Programs

Mathematica

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

PARI

seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 27 2024

Extensions

a(26) onwards from Andrew Howroyd, Jul 27 2024