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Nonassociative Product

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

F(n,k)=F(n-1,k)+F(n-1,k-1)
(1)
=(n+k-2; k)-(n+k-1; k-1),
(2)

where (n; k) is a binomial coefficient. The number of n-products in a nonassociative algebra is

F(n)=C_n=sum_(j=0)^(n-2)F(n,j)=((2n-2)!)/(n!(n-1)!),
(3)

where C_n is a Catalan number, 1, 1, 2, 5, 14, 42, 132, ... (OEIS A000108).

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References

Niven, I. M. Mathematics of Choice: Or, How to Count Without Counting. Washington, DC: Math. Assoc. Amer., pp. 140-152, 1965.Sloane, N. J. A. Sequence A000108/M1459 in "The On-Line Encyclopedia of Integer Sequences."

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Nonassociative Product

Cite this as:

Weisstein, Eric W. "Nonassociative Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonassociativeProduct.html

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