|
| |
|
|
A180171
|
|
Triangle read by rows: R(n,k) is the number of k-reverses of n.
|
|
4
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 4, 10, 5, 1, 1, 6, 9, 12, 15, 6, 1, 1, 7, 9, 19, 15, 21, 7, 1, 1, 8, 10, 24, 30, 20, 28, 8, 1, 1, 9, 12, 36, 26, 54, 28, 36, 9, 1, 1, 10, 15, 40, 50, 60, 70, 40, 45, 10, 1, 1, 11, 13, 53, 50, 108, 70, 106, 39, 55, 11, 1, 1, 12, 18, 60, 75, 120
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
COMMENTS
|
A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
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.
For example 114 is a 3-reverse of 6 since its set of cyclic equivalents {114,411,141} contains its reverse 411. But 123 is not a 3-reverse of 6 since its set of cyclic equivalents {123,312,231} does not contain its reverse 321.
|
|
|
REFERENCES
|
John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
|
|
|
LINKS
|
|
|
|
FORMULA
|
For proofs of these formulae, see the links.
R(n,k) = Sum_{d|gcd(n,k)} phi^{(-1)}(d)*(k/d)*A119963(n/d, k/d), where phi^{(-1)}(d) = A023900(d) is the Dirichlet inverse function of Euler's totient function.
G.f.: Sum_{s >= 1} phi^{(-1)}(s)*g(x^s, y^s), where phi^{(-1)}(s) = A023900(s) and g(x,y) = (x*y+x+1)*(x*y-x+1)*(x+1)*x*y/(x^2*y^2+x^2-1)^2.
(End)
|
|
|
EXAMPLE
|
The triangle begins
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 4 10 5 1
1 6 9 12 15 6 1
1 7 9 19 15 21 7 1
1 8 10 24 30 20 28 8 1
1 9 12 36 26 54 28 36 9 1
For example row 8 is 1 7 9 19 15 21 7 1
We have R(8,3)=9 because there are 9 3-reverses of 8. In classes: {116,611,161} {224,422,242}, and {233,323,332}.
We have R(8,6)=21 because all 21 6-compositions of 8 are 6-reverses of 8.
|
|
|
MATHEMATICA
|
f[n_Integer, k_Integer] := Block[{c = 0, j = 1, ip = IntegerPartitions[n, {k}]}, lmt = 1 + Length@ ip; While[j < lmt, c += g[ ip[[j]]]; j++ ]; c]; g[lst_List] := Block[{c = 0, len = Length@ lst, per = Permutations@ lst}, While[ Length@ per > 0, rl = Union[ RotateLeft[ per[[1]], # ] & /@ Range@ len]; If[ MemberQ[rl, Reverse@ per[[1]]], c += Length@ rl]; per = Complement[ per, rl]]; c]; Table[ f[n, k], {n, 13}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 25 2010 *)
|
|
|
PROG
|
p(n, k) = binomial((n-k%2)\2, k\2);
AR(n, k) = k*sumdiv(gcd(n, k), d, moebius(d) * p(n/d, k/d));
T(n, k) = sumdiv(gcd(n, k), d, AR(n/d, k/d));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Oct 08 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
