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Number of permutations p of [n] such that the n-1 sums p(i)+p(i+1) (i=1,2,...n-1) are all distinct.
3

%I #14 Jul 06 2021 16:11:58

%S 1,1,2,6,16,54,232,1132,6024,36262,242080,1775316,14135584,122077832,

%T 1131066448,11230979624,118638940864

%N Number of permutations p of [n] such that the n-1 sums p(i)+p(i+1) (i=1,2,...n-1) are all distinct.

%e For a(4), we have 2+3=1+4, so a(4)=4! - (23)(14) - (14)(32) = 24 - 4 - 4 = 16.

%e For a(5), we have 1+4=2+3, 1+5=2+4, 2+5=3+4 to avoid.

%o (PARI) { for (i=3,9, v=vector(i-1); c=0; for (j=1,i!, x=numtoperm(i,j); for (k=1,i-1, v[k]=x[k]+x[k+1]); fl=0; v=vecsort(v); for (z=1,i-2,if (v[z]==v[z+1],fl=1;break)); if (fl==0,c++)); print1(","c)) }

%K hard,more,nonn

%O 0,3

%A _Jon Perry_, Feb 23 2004

%E a(10)-a(11) from _John W. Layman_, Mar 10 2004

%E More terms from _R. H. Hardin_, Nov 11 2008

%E Definition corrected by _Emeric Deutsch_, Nov 28 2008

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 06 2021