A140257 Number of permutations p of order n such that the system of congruences x == i (mod p(i)), i=1..n, is solvable.

1, 2, 6, 8, 48, 24, 216, 120, 240, 128, 2544, 336, 11520, 3168, 1536, 480, 23616, 2592

Offset: 1

Author

Max Alekseyev, May 16 2008, May 17 2008

Keywords

• nonn

Comments

The system of congruences x == i (mod p(i)), i=1..n, is solvable if and only if for every pair of indices i,j=1..n, gcd(p(i),p(j)) divides (i-j).

Note that if the system is solvable for a permutation p=(p(1),p(2),...,p(n)), then it is solvable for reversed permutation (p(n),p(n-1),...,p(1)). Also, any two primes q1, q2 greater than n/2 in p can be exchanged without affecting the system solvability. Therefore for n>1, a(n) is divisible by 2*(A000720(n)-A000720(n/2))!.

Programs

PARI

{ allper(n,i) = local(b); if(i>n,r++;return); p[i]=0; while(p[i]m,next); b=0; for(j=1,i-1, if((i-j)%gcd(p[i],p[j]),b=1;break)); if(b,next); P[p[i]]=1; if(q[p[i]]==m,m++;allper(n,i+1);m--,allper(n,i+1)); P[p[i]]=0) } { a(n) = P=p=q=vector(n); for(i=1,n,if(isprime(i),q[i]=primepi(i))); m=primepi(n\2)+1; r=0; allper(n,1); r*(primepi(n)-primepi(n\2))! }