A374686 - OEIS

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

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OFFSET

0,3

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.

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).

EXAMPLE

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)

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A074045 A073776 A129853 * A095982 A095090 A061947

Adjacent sequences: A374683 A374684 A374685 * A374687 A374688 A374689

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 27 2024

EXTENSIONS

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

STATUS

approved