OFFSET
1,12
COMMENTS
A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of at least two smaller compositions.
Two k-compositions are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
The reverse of a k-composition is the k-composition obtained by writing its parts in reverse. For example, the reverse of 123 is 321.
A k-reverse of n is a k-composition of n which is cyclically equivalent to its reverse. And an aperiodic k-reverse of n is a k-reverse of n which is aperiodic.
For example, 114 is an aperiodic 3-reverse of 6 since it is aperiodic and its set of cyclic equivalents {114,411,141} contains its reverse 411.
But 123 is not an aperiodic 3-reverse of 6 since, even though it is aperiodic, its set of cyclic equivalents {123,312,231} does not contain its reverse 321.
Let AR(n,k) denote the number of aperiodic k-reverses of n, then sequence A180279 is the 'AR(n,k)' triangle read by rows.
For the above sequence we count the aperiodic k-reverses of n up to cyclic equivalence, ARE(n,k), in other words, the number of equivalence classes under cyclic permutation which contain at least one aperiodic k-reverse of n.
REFERENCES
John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
ARE(n,k) = Sum_{d|gcd(n,k)} mu(d) * A119963(n/d,k/d). - Andrew Howroyd, Oct 07 2017
EXAMPLE
The triangle begins
1
1 0
1 1 0
1 1 1 0
1 2 2 1 0
1 2 1 2 1 0
1 3 3 3 3 1 0
1 3 3 4 3 3 1 0
1 4 3 6 6 3 4 1 0
1 4 4 8 5 8 4 4 1 0
For example, row 8 is 1 3 3 4 3 3 1 0.
We have ARE(8,3)=3 because there are 9 aperiodic 3-reverses of 8 in the 3 classes {116,611,161}, {224,422,242}, and {233,323,332}, and so there are ARE(8,3)=3 aperiodic 3-reverses of 8 up to cyclic equivalence.
We have ARE(8,6)=3 because there are 3 aperiodic 6-reverses of 8 up to cyclic equivalence. The representatives of the 3 classes are 111113, 111122, and 111212.
MATHEMATICA
Table[DivisorSum[GCD[n, k], MoebiusMu[#]*Binomial[Floor[((n/#) - Boole[OddQ[k/#]])/2], Floor[k/(2 #)]] &], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021 *)
PROG
(PARI) \\ here RE(n, k) is A119963(n, k).
RE(n, k) = binomial((n-k%2)\2, k\2);
T(n, k) = sumdiv(gcd(n, k), d, moebius(d)*RE(n/d, k/d));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Oct 07 2017
CROSSREFS
As mentioned above, if we don't count the classes, but rather the elements in the classes, we get sequence A180279.
If we remove the aperiodic requirement, see sequence A119963.
KEYWORD
nonn,tabl
AUTHOR
John P. McSorley, Sep 03 2010
EXTENSIONS
Terms a(56) and beyond from Andrew Howroyd, Oct 07 2017
STATUS
approved