[seqfan] A combinatorial problem

Vladimir Shevelev shevelev at bgu.ac.il
Fri Jan 14 13:19:13 CET 2011

Dear SeqFans,
I ask anyone to extend a sequence which is connected with the following modification of the menage problem. A well known mathematician N found himself with his wife among the guests, which were
n(>=3) married couples. After seating the ladies on every other chair at a circular table, N was the first offered to choose an arbitrary chair but not side by side with his wife. For which values of n the number of ways of seating of other men ( under the condition that no husband is beside his wife) does not depend on how far N takes his seat from his wife?
The first terms of this sequence are 3,4,6.  I proved that the problem reduces to description the values of n>=3 for which, for every r=1,...,n, we have
where B=Sum{i=0,...,k}C(2r-i-4, i)*C(2n-2r-k+i+2, k-i), i.e., for such an n, B does not depend on r  (here C-binomial coefficients).
In addition, I proved that A000179(n)/(n-2) is integer, if n has the form 2^t+2 ( and I conjecture that here one can write "iff").
E.g., if n=3, then, for every r,  if k=0, then B=1; if k=1, then B=2; if k=2, then B=1. Thus

 Shevelev Vladimir‎

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