A374760 - OEIS

A374760

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

1, 1, 2, 3, 4, 6, 8, 11, 15, 21, 28, 38, 52, 70, 95, 129, 173, 234, 318, 428, 579, 784, 1059, 1433, 1942, 2630, 3564, 4835, 6559, 8902, 12094, 16432, 22340, 30392, 41356, 56304, 76692, 104499, 142448, 194264, 265015, 361664, 493749, 674278, 921113, 1258717

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OFFSET

0,3

COMMENTS

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

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).

FORMULA

G.f.: 1 + Sum_{k>=1} -1 + 1/(1 - x^k*Product_{j=1..k-1} (1 + x^j)). - Andrew Howroyd, Jul 31 2024

EXAMPLE

The composition (3,3,2,1,3,2,1) has strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so is counted under a(15).

The a(0) = 1 through a(8) = 15 compositions:

() (1) (2) (3) (4) (5) (6) (7) (8)

(11) (21) (22) (32) (33) (43) (44)

(111) (31) (41) (42) (52) (53)

(1111) (212) (51) (61) (62)

(221) (222) (313) (71)

(11111) (321) (331) (323)

(2121) (421) (332)

(111111) (2122) (431)

(2212) (521)

(2221) (2222)

(1111111) (3131)

(21212)

(21221)

(22121)

(11111111)

MATHEMATICA

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

PROG

(PARI) seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=1, min(n-k, k-1), 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 31 2024

CROSSREFS

For partitions instead of compositions we have A034296.

The weak version is A374742, ranks A374744.

The opposite version is A374686, ranks A374685.

The weak opposite version is A374631, ranks A374633.

Ranked by A374759.

Other types of runs (instead of strictly decreasing):

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

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

Other types of run-leaders (instead of identical):

- For distinct leaders we have A374761, ranks A374767.

- For strictly increasing leaders we have A374762.

- For strictly decreasing leaders we have A374763.

- For weakly increasing leaders we have A374764.

- For weakly decreasing leaders we have A374765.

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.

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

Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374640, A374761.

Sequence in context: A188917 A046935 A241334 * A095814 A006683 A014213

Adjacent sequences: A374757 A374758 A374759 * A374761 A374762 A374763

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 29 2024

EXTENSIONS

a(24) onwards from Andrew Howroyd, Jul 31 2024

STATUS

approved