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Nonassociative Product

The number of nonassociative $n$-products with $k$ elements preceding the rightmost left parameter is

$\displaystyle F(n,k)$ $\textstyle =$ $\displaystyle F(n-1,k)+F(n-1,k-1)$  
  $\textstyle =$ $\displaystyle {n+k-2\choose k}-{n+k-1\choose k-1},$  

where ${n\choose k}$ is a Binomial Coefficient. The number of $n$-products in a nonassociative algebra is

F(n)=\sum_{j=0}^{n-2} F(n,j)={(2n-2)!\over n!(n-1)!}.


Niven, I. M. Mathematics of Choice: Or, How to Count Without Counting. Washington, DC: Math. Assoc. Amer., pp. 140-152, 1965.

© 1996-9 Eric W. Weisstein