A122158 Smallest positive number of "triangular" shuffles of n(n+1)/2 cards needed to restore them to their original order.
1, 2, 4, 20, 18, 12, 126, 33, 204, 1638, 1968, 2010, 504, 17043, 240, 222870, 4084080, 12462, 81396, 4200, 41790, 24254, 56560, 1377090, 30669496, 57420, 249480, 93840, 696010, 40680, 4627497204, 2122260, 4774320, 347190, 83800, 103458, 5017446420, 1686300
Offset: 1
Keywords
Examples
For n=3, successive shuffles give: 1.......4.......5.......3.......1 2.3.....2.1.....2.4.....2.5.....2.3 4.5.6...5.3.6...3.1.6...1.4.6...4.5.6 returning the deck of 6 cards to its original order in 4 shuffles. Thus a(3)=4.
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..500
Programs
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PARI
Perm(n)={concat(Vecrev(vector(n,i,vector(i,j,n+1-i+(n-j+1)*(n-j)/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
Extensions
Terms a(17) and beyond from Andrew Howroyd, Nov 12 2017
Comments