A180424 "ARE(n,k)" triangle read by rows. ARE(n,k) is the number of aperiodic k-reverses of n up to cyclic equivalence.

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, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 0, 1, 5, 4, 12, 10, 17, 10, 12, 4, 5, 1, 0

Offset: 1

Author

John P. McSorley, Sep 03 2010

Keywords

• nonn
• tabl

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.

Examples

			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.
		

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

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.

The row sums of the "ARE(n, k)" triangle above give sequence A056493 (except for the first term). See also sequence A056498.

Programs

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 *)

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

Formula

ARE(n,k) = Sum_{d|gcd(n,k)} mu(d) * A119963(n/d,k/d). - Andrew Howroyd, Oct 07 2017

Extensions

Terms a(56) and beyond from Andrew Howroyd, Oct 07 2017