|
| |
|
|
A119963
|
|
Triangle T(n,k), 0<=k<=n, read by rows,with T(2n,2k)=T(2n+1,2k)=T(2n+1,2k+1)=T(2n+2,2k+1)=binomial(n,k).
|
|
4
| |
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 1, 5, 4, 10, 6, 10, 4, 5, 1, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,13
|
|
|
COMMENTS
| Contribution from John P. McSorley (mcsorley60(AT)hotmail.com), Aug 24 2010: (Start)
A combinatorial interpretation of this triangle is as follows:
Ignore the first column of 1's of the above triangle and the call the (n,k)
entry of the new triangle formed RE(n,k).
Hence row 8 of the `RE(n,k)' triangle is 1 4 3 6 3 4 1 1.
Now see sequence A180171 for the definition of a k-reverse of n.
Briefly, a k-reverse of n is a k-composition of n which is cyclically equivalent to its reverse.
Sequence A180171 is the `R(n,k)' triangle read by rows where R(n,k) is the total number of k-reverses of n.
Then RE(n,k) is the number of k-reverses of n upto cyclic equivalence.
In sequence A180171 we have R(8,3)=9 because there are 9 3-reverses of 8.
In cyclically equivalent classes: {116,611,161} {224,422,242}, and {233,323,332};
since there are 3 such classes we have RE(8,3)=3.
Similarly, in A180171, we have R(8,6)=21 because all 21 6-compositions of 8
are 6-reverses of 8, but they come in 4 cyclically equivalent classes
(with representatives 111113, 111122, 111212, and 112112) hence RE(8,6)=4.
There is another (equivalent) interpretation for RE(n,k) involving k-subsets of Z_n,
the integers modulo n, and the multiplier -1. See the McSorley/Schoen paper below for more details.
In this case it is convenient to count k-subsets upto dihedral equivalence, rather than cyclic equivalence.
The counts are the same. The row sums of the `RE(n,k)' triangle give sequence A052955.
(End)
|
|
|
REFERENCES
| John P.McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [From John P. McSorley (mcsorley60(AT)hotmail.com), Aug 24 2010]
John P.McSorley and Alan H. Schoen: On k-Ovals and (n, k, lambda)-Cyclic Difference Sets, and Related Topics. Preprint, 2010. [From John P. McSorley (mcsorley60(AT)hotmail.com), Aug 24 2010]
|
|
|
EXAMPLE
| Triangle begins:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 2, 1, 1;
1, 1, 2, 2, 1, 1;
1, 1, 3, 2, 3, 1, 1;
1, 1, 3, 3, 3, 3, 1, 1;
1, 1, 4, 3, 6, 3, 4, 1, 1,;
|
|
|
CROSSREFS
| Cf. A007318, A051159.
The row sums of the T(n,k) triangle give sequence A029744 whose terms are 1 more than the terms of sequence A052955 (row sums of RE(n,k) triangle). See sequence A029744 where there is a reference to necklaces relevant to the combinatorial interpretation and the McSorley and McSorley/Schoen papers given here. [From John P McSorley (mcsorley60(AT)hotmail.com), Aug 31 2010]
Sequence in context: A055801 A155050 A140356 * A057790 A052307 A067059
Adjacent sequences: A119960 A119961 A119962 * A119964 A119965 A119966
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006
|
|
|
EXTENSIONS
| Corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 20 2010
|
| |
|
|
