A105272 Array T(n,k) (k >= 1, n >= k) read by antidiagonals (see definition in Comments lines).

1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 4, 2, 2, 1, 3, 6, 4, 2, 2, 1, 3, 6, 7, 4, 2, 2, 1, 6, 4, 3, 7, 4, 2, 2, 1, 6, 4, 3, 15, 14, 4, 2, 2, 1, 10, 4, 8, 5, 6, 14, 4, 2, 2, 1, 10, 21, 10, 5, 10, 6, 14, 4, 2, 2, 1, 12, 3, 6, 12, 12, 12, 6, 14, 4, 2, 2

Offset

1,3

Comments

T(n,k) is the order of the permutation p of [1,...,n] defined as follows:

Write F={1,2,3,....,n}.

Place F into a "window" of width k, where k <= n. That is, write out the elements from left to right, up to down, with k elements per line.

Produce a new set F' by traversing the set according to the following algorithm, adding elements to F' as they are traversed in F.

Traversal algorithm:

1) Start at the upper right hand element.

2) If there is an element below the current one

then

A) go to it

B) go back to step 2

3) Otherwise, if there is a column to the left of the current one, then

A) go to it

B) go back to step 2

4) End

Then p is the permutation that sends F to F'.

Links

Robert Price, Table of n, a(n) for n = 1..1275

Samuel Minter, Abulsme function information and definition

Example

To find T(12,5):
Start with F = { A B C D E F G H I J K L } with a window of widhth 5:
A B C D E
F G H I J
K L
Now let's traverse that and construct our new set
Upper right is E so add it to our new set:
{ E ....
We can go down so we do so and get J
{ E J .....
Now we can't go down so go to the top of the column to the left and get D
{ E J D .....
Eventually we will get:
F' = { E J D I C H B G L A F K }
The permutation p that sends F to F' is a single cycle of length 12, so T(12,5) = 12.
Array begins:
k = 1: 1,1,1,1,1,1,1,1,1,1,... (A000012)
k = 2: 2,2,4,4,3,3,6,6,10,10,... (A024222)
k = 3: 2,2,4,6,6,4,4,4,21,3,... (A118960)
k = 4: 2,2,4,7,3,3,8,10,6,6,... (A120280)
k = 5: 2,2,4,7,15,5,5,12,40,45,... (A120363)
k = 6: 2,2,4,14,6,10,12,12,7,15,... (A120654)
k = 7: 2,2,4,14,6,12,30,4,4,20,... (A121514)
k = 8: 2,2,4,14,6,13,13,24,8,8,...
k = 9: 2,2,4,14,6,13,15,15,63,9,...
k = 10: 2,2,4,14,6,13,16,10,18,12,...
... (Rows converge to A121526)

Mathematica

T[1] = ConstantArray[1, 75];
For[k = 2, k <= 20, k++,
  T[k] = Table[f = Range[n]; fp = {};
    For[col = k, col > 0, col--,
     For[row = 0, col + row*k <= n, row++,
      AppendTo[fp, f[[col + row*k]]]]];
    LCM @@ Length /@ First[FindPermutation[f, fp]], {n, k, 75}]];
A105272 = {};
For[i = 1, i <= 20, i++,
  For[j = i, j >= 1, j--,
   AppendTo[A105272, T[i - j + 1][[j]]]]];
A105272 (* Robert Price, Aug 26 2019 *)

Prog

(C)
int abulsme(int l, int s)
{
    long int t[30000], m[30000], c[30000], b[30000];
    long int k, i, n, j, z, u, q, g;
    for (t[1] = s, k = 2; k <= l; k++)
    {
        m[k] = (t[k - 1] + s - l + abs(t[k - 1] + s - l)) / (2 * abs(t[k - 1] + s - l - 1) + 2);
        t[k] = ((t[k - 1] - m[k]) % (s * m[k] + 2 * l * abs(m[k] - 1))) + s * abs(m[k] - 1);
    }
    for (i = 1; i <= l; b[i] = 0, i++)
        ;
    for (n = 0, i = 1; i <= l; i++)
    {
        if (!b[i])
        {
            j = i;
            k = 0;
            do
            {
                j = t[j];
                b[j] = 1;
                k++;
            } while (j != i);
            u = 1;
            z = 1;
            if (i > 1)
            {
                do
                {
                    if (c[z] == k)
                    {
                        u = 0;
                    }
                    z++;
                } while (!((z > n) || (!u)));
            }
            if (u)
            {
                n++;
                c[n] = k;
            }
        }
        for (q = c[1], g = q, z = 1; z < n; z++, g = q)
        {
            for (0; q % c[z + 1]; q += g)
                ;
        }
    }
    return g;
}

Crossrefs

Sequence in context: A363345 A344651 A308558 * A060438 A106190 A029290

Adjacent sequences: A105269 A105270 A105271 * A105273 A105274 A105275

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 10 2008, based on email from Samuel Minter (abulsme(AT)abulsme.com0, May 08 2008

Extensions

a(46)-a(78) from Robert Price, Aug 26 2019

Status

approved