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A374632
Number of integer compositions of n whose leaders of weakly increasing runs are distinct.
43
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
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.
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.
Sequence in context: A335471 A168043 A374764 * A114832 A239553 A319255
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 23 2024
STATUS
approved