OFFSET
0,3
COMMENTS
These are finite sequences q of positive integers summing to n such that q(i) >= q(i+2) for all possible i.
The strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?
Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
EXAMPLE
The a(1) = 1 through a(6) = 20 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(121) (41) (42)
(211) (131) (51)
(1111) (212) (141)
(221) (222)
(311) (231)
(1211) (312)
(2111) (321)
(11111) (411)
(1212)
(1311)
(2121)
(2211)
(3111)
(12111)
(21111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], GreaterEqual@@Plus@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}]
PROG
(PARI) seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(1+sum(k=1, n, polcoef(p, k, y)*(polcoef(p, k-1, y) + polcoef(p, k, y))))} \\ Andrew Howroyd, Mar 24 2021
CROSSREFS
The even-length case is A114921.
The version with alternating parts equal is A342527.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2021
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Mar 24 2021
STATUS
approved