Roman Coefficient

A generalization of the Binomial Coefficient whose Notation was suggested by Knuth,

$$\left\lfloor{\matrix{n\cr k\cr}}\right\rceil ={\left\lfloor{... ...\left\lfloor{k}\right\rceil !\left\lfloor{n-k}\right\rceil !}.$$ (1)

The above expression is read ``Roman $n$ choose $k$.'' Whenever the Binomial Coefficient is defined (i.e., $n\geq k\geq 0$ or $k\geq 0>n$), the Roman coefficient agrees with it. However, the Roman coefficients are defined for values for which the Binomial Coefficients are not, e.g.,

$$\left\lfloor{\begin{array}{c}n\\ -1\end{array}}\right\rceil = {1\over\left\lfloor{n+1}\right\rceil }$$ (2)

$$\left\lfloor{\begin{array}{c}0\\ k\end{array}}\right\rceil = {(-1)^{k+(k>0)}\over\left\lfloor{k}\right\rceil },$$ (3)

where

$$n<0\equiv\cases{ 1 & for $n<0$\cr 0 & for $n\geq 0$.\cr}$$ (4)

The Roman coefficients also satisfy properties like those of the Binomial Coefficient,

$$\left\lfloor{\matrix{n\cr k\cr}}\right\rceil =\left\lfloor{\matrix{n\cr n-k\cr}}\right\rceil$$ (5)

$$\left\lfloor{\matrix{n\cr k\cr}}\right\rceil \left\lfloor{\m... ...right\rceil \left\lfloor{\matrix{n-r\cr k-r\cr}}\right\rceil ,$$ (6)

an analog of Pascal's Formula

$$\left\lfloor{\matrix{n\cr k\cr}}\right\rceil =\left\lfloor{\... ...ight\rceil +\left\lfloor{\matrix{n-1\cr k-1\cr}}\right\rceil ,$$ (7)

and a curious rotation/reflection law due to Knuth

$$(-1)^{k+(k>0)}\left\lfloor{\matrix{-n\cr k-1\cr}}\right\rceil =(-1)^{n+(n>0)}\left\lfloor{\matrix{-k\cr n-1\cr}}\right\rceil$$ (8)

(Roman 1992).

See also Binomial Coefficient, Roman Factorial

References

Roman, S. ``The Logarithmic Binomial Formula.'' Amer. Math. Monthly 99, 641-648, 1992.