A093057 Triangle T(j,k) read by rows, where T(j,k) = number of matrix elements remaining at fixed position in the in-situ transposition of a rectangular j X k matrix (singleton cycles).

0, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 0, 0, 0, 4, 0, 1, 2, 1, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 1, 0, 3, 0, 1, 0, 7, 0, 0, 2, 0, 0, 2, 0, 0, 8, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 1, 0

Offset

1,6

Comments

Elements (1,1) and (j,k) which always remain at their old position are not counted. See A093055 for details of storage, another example, references and links.

Links

Table of n, a(n) for n=1..104.

Formula

T(j, k)=gcd(j-1, k-1)-1.

Example

a(8)=T(3,5)=1 because there is one fixed element at position 8 in the transposition from
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15) ->
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15). The fixed first and last elements 1 and 15 are not counted.

Crossrefs

Cf. A093055 number of non-singleton cycles, A093056 length of longest cycle, A050873 GCD(u, v).

Sequence in context: A357882 A103775 A331594 * A065334 A356945 A162590

Adjacent sequences: A093054 A093055 A093056 * A093058 A093059 A093060

Keyword

nonn,tabl

Author

Hugo Pfoertner, Mar 22 2004

Status

approved