Message From Richard Stanley

From rstan at math dot mit dot edu  Mon Jan  2 21:36:03 2006

Dear Richard, Margie, and Doron,

Happy New Year! I came across Doron's journal entry "I Am Sorry,
Richard Ehrenborg and Margie Readdy, About Your Two Conjectures, But
One Is FAMOUS, While The Other Is FALSE," which inspired me to look
for a generating function for the number a(m) of up-down involutions
of 1,2,...,m. 

The result is as follows.

For m=2n+1 :

 Sum_{n=0..infinity} a(2n+1)x^(2n+1) =
 Sum_{i,j=0..infinity) arctan(x)^(2i+1) (log((1+x^2)/(1-x^2)))^j
 E(2i+2j+1)/(2i+1)!j!4^j   .

For m=2n:

 Sum_{n=0..infinity} a(2n)x^(2n) =
  (1-x^4)^(-1/4) Sum_{i,j=0..infinity) arctan(x)^(2i) 
     (log((1+x^2)/(1-x^2)))^j E(2i+2j)/(2i)!j!4^j .


Here E(2i+2j+1) is an Euler number. 

Best regards,
Richard