A374632 - OEIS

A374632

Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

1, 1, 2, 4, 7, 13, 23, 40, 69, 119, 200, 335, 557, 917, 1499, 2433, 3920, 6280, 10004, 15837, 24946, 39087, 60952, 94606, 146203, 224957

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OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..25.

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

EXAMPLE

The composition (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).

The a(0) = 1 through a(5) = 13 compositions:

() (1) (2) (3) (4) (5)

(11) (12) (13) (14)

(21) (22) (23)

(111) (31) (32)

(112) (41)

(211) (113)

(1111) (122)

(212)

(221)

(311)

(1112)

(2111)

(11111)

MATHEMATICA

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

CROSSREFS

Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).

Types of runs (instead of weakly increasing):

- For leaders of constant runs we have A274174, ranks A374249.

- For leaders of anti-runs we have A374518, ranks A374638.

- For leaders of strictly increasing runs we have A374687, ranks A374698.

- For leaders of weakly decreasing runs we have A374743, ranks A335467.

- For leaders of strictly decreasing runs we have A374761, ranks A374767.

Types of run-leaders (instead of distinct):

- For strictly decreasing leaders we appear to have A188920.

- For weakly decreasing leaders we appear to have A189076.

- For identical leaders we have A374631.

- For weakly increasing leaders we have A374635.

- For strictly increasing leaders we have A374634.

A003242 counts anti-run compositions.

A011782 counts compositions.

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

A335456 counts patterns matched by compositions.

A374637 counts compositions by sum of leaders of weakly increasing runs.

Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.

Sequence in context: A335471 A168043 A374764 * A114832 A239553 A319255

Adjacent sequences: A374629 A374630 A374631 * A374633 A374634 A374635

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 23 2024

STATUS

approved