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A329749
Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.
3
1, 1, 0, 2, 3, 5, 11, 23, 40, 80, 180, 344, 661, 1321, 2657, 5268, 10481, 20903, 41572, 82734, 164998, 328304, 654510, 1305421, 2598811, 5182174, 10332978, 20594318, 41066611, 81897091, 163309679, 325707492, 649648912, 1295827380, 2584941276, 5156774487
OFFSET
0,4
COMMENTS
A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.
EXAMPLE
The a(0) = 1 through a(6) = 11 compositions (empty column not shown):
() (1) (1,2) (1,1,2) (1,2,2) (1,2,3)
(2,1) (1,2,1) (2,1,2) (1,3,2)
(2,1,1) (2,2,1) (2,1,3)
(1,1,2,1) (2,3,1)
(1,2,1,1) (3,1,2)
(3,2,1)
(1,2,1,2)
(1,2,2,1)
(2,1,1,2)
(2,1,2,1)
(1,1,2,1,1)
MATHEMATICA
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], normQ[#]&&normQ[Length/@Split[#]]&]], {n, 0, 10}]
CROSSREFS
Looking at multiplicities instead of run-lengths gives A329748.
The non-complete version is A329766.
Complete compositions are A107429.
Sequence in context: A038905 A019405 A079848 * A237810 A073434 A109551
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 21 2019
EXTENSIONS
a(21)-a(35) from Alois P. Heinz, Jul 06 2020
STATUS
approved