Definition:Hat-Check Triangle

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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 $n$:


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

This sequence is A098825 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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


Also see


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