OFFSET
0,4
COMMENTS
These are finite odd-length sequences q of distinct positive integers summing to n such that q(i) > q(i+2) for all possible i.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{k>=0} binomial(k,floor(k/2)) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021
EXAMPLE
The a(1) = 1 through a(8) = 13 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7)
(2,1) (3,1) (2,3) (2,4) (2,5) (2,6)
(3,2) (4,2) (3,4) (3,5)
(4,1) (5,1) (4,3) (5,3)
(2,3,1) (5,2) (6,2)
(3,1,2) (6,1) (7,1)
(3,2,1) (2,4,1) (2,5,1)
(4,1,2) (3,4,1)
(4,2,1) (4,1,3)
(4,3,1)
(5,1,2)
(5,2,1)
MATHEMATICA
ici[q_]:=And@@Table[q[[i]]>q[[i+2]], {i, Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], ici]], {n, 0, 15}]
PROG
(PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, binomial(k, k\2) * polcoef(p, k, y)))} \\ Andrew Howroyd, Apr 16 2021
CROSSREFS
The non-strict odd-length case is A001522.
Strict compositions in general are counted by A032020
The non-strict even-length case is A064428.
The case of reversed partitions is A065033.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A027193 counts odd-length compositions.
A034008 counts even-length compositions.
A064391 counts partitions by crank.
A064410 counts partitions of crank 0.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
A325548 counts compositions with strictly decreasing differences.
A342194 counts strict compositions with equal differences.
A342527 counts compositions with alternating parts equal.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 01 2021
STATUS
approved