Definition:Hat-Check Triangle

Definition
The Hat-Check Triangle is an array formed by the product of binomial coefficients with the subfactorial which produces the hat-check distribution for any value of $n$:


 * $\begin{array}{r|rrrrrrrrrr}

n & !0 \binom n 0 & !1 \binom n 1 & !2 \binom n 2 & !3 \binom n 3 & !4 \binom n 4 & !5 \binom n 5 & !6 \binom n 6 & !7 \binom n 7 & !8 \binom n 8 & !9 \binom n 9 & !10 \binom n {10} \\ \hline 0 & 1 &  0 &  0 &  0  &  0   &   0   &   0   &   0    &   0    &  0  &  0 \\ 1  & 1 &  0 &  0 &  0  &  0   &   0   &   0   &   0    &   0    &  0  &  0 \\ 2  & 1 &  0 &  1 &  0  &  0   &   0   &   0   &   0    &   0    &  0  &  0 \\ 3  & 1 &  0 &  3 &  2  &  0   &   0   &   0   &   0    &   0    &  0  &  0 \\ 4  & 1 &  0 &  6 &  8  &  9   &   0   &   0   &   0    &   0    &  0  &  0 \\ 5  & 1 &  0 & 10 & 20  & 45   &  44   &   0   &   0    &   0    &  0  &  0 \\ 6  & 1 &  0 & 15 & 40  & 135  &  264  & 265   &   0    &   0    &  0  &  0 \\ 7  & 1 &  0 & 21 & 70  & 315  &  924  & 1855  & 1854   &   0    &  0  &  0 \\ 8  & 1 &  0 & 28 & 112 & 630  & 2464  & 7420  & 14832  & 14833  &  0  &  0 \\ 9  & 1 &  0 & 36 & 168 & 1134 & 5544  & 22260 & 66744  & 133497 &  133496  &  0 \\ 10 & 1 &  0 & 45 & 240 & 1890 & 11088 & 55650 & 222480 & 667485 &  1334960 &  1334961 \\ \end{array}$

Thus the entry in row $n$ and column $k$ contains the product of the binomial coefficient $\dbinom n k$ with the subfactorial $!k$.

Recall from the hat-check distribution:

$\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$

Therefore:

Also see

 * Definition:Hat-Check Distribution
 * Definition:Pascal's Triangle
 * Definition:Stirling's Triangles