A048782 Smallest positive number of "triangular" shuffles of n(n+1)/2 cards needed to restore them to their original order.

1, 1, 2, 6, 10, 84, 70, 24, 36, 330, 210, 288, 1428, 252, 10080, 13680, 1260, 19320, 2160, 22440, 1692152, 50400, 140760, 3071880, 284088, 608753236, 617760, 15600, 35287560, 214138080, 30240, 1866600, 58333800, 8637552, 4034916600

Offset

1,3

Comments

Lay out cards in a triangular array from left to right in rows with 1 card in row 1 (the top row), 2 cards in row 2, etc., then pick up by columns, in order from the top of the column to the bottom, first from column 1, then column 2, etc. See A122158 for analogous results for a different triangular shuffle.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

For n=3, successive shuffles give:
1.......1.......1
2.3.....2.4.....2.3
4.5.6...3.5.6...4.5.6,
returning the deck of 6 cards to its original order in 2 shuffles. Thus a(3)=2.

Prog

(PARI)
Perm(n)={concat(vector(n, i, vectorsmall(i, j, i+n*(j-1)-j*(j-1)/2)))}
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
CyclePoly(n, x)={my(v=Perm(n), q=0); for(i=1, #v, my(l=Follow(i, j->v[j])); if(l, q+=x^l)); q}
a(n)={my(q=CyclePoly(n, x), m=1); for(i=1, poldegree(q), if(polcoeff(q, i), m=lcm(m, i))); m} \\ Andrew Howroyd, Nov 12 2017

Crossrefs

Cf. A121052, A122158.

Sequence in context: A093880 A080397 A322756 * A358739 A083458 A124621

Adjacent sequences: A048779 A048780 A048781 * A048783 A048784 A048785

Keyword

nonn

Author

John W. Layman, Jul 14 1999, Aug 22 2006

Extensions

Edited by R. J. Mathar, Aug 02 2008

Status

approved