At MIT all departments have numbers, and math is 18. Last year MIT math majors produced a tee shirt that said ${i\choose 18}$ ("I choose 8"18") on the front, and on the back $$ \frac{34376687+1499084559i}{14485008384}. $$ With the more natural denominator $18!$ this is $$ \frac{15194495654000+662595375078000i}{18!}. $$ This suggests the question: for any $n\geq 1$ find a "nice" combinatorial interpretation of the real and imaginary parts of $i(i-1)(i-2)\cdots (i-n+1)=f_n+ig_n$. It is easy to express $f_n$ and $g_n$ as certain alternating sums of Stirling numbers of the first kind, but I don't consider this "nice." The $g_n$'s seem to alternate in sign beginning with $n=5$. The $f_n$'s alternate in sign up to $n=17$ and then seem to alternate in sign beginning with $n=18$. It is curious that $i(i-1)(i-2)(i-3)=-10$, a real number. One could ask the same question with $i$ replaced by any Gaussian integer $a+bi$. One can also ask about the asymptotic rate of growth of $f_n$ and $g_n$. Clearly $f_n^2+g_n^2\sim C\cdot (n-1)!^2$, so one would expect $f_n$ and $g_n$ to be roughly of the size of $(n-1)!$.
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Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$At MIT all departments have numbers, and math is 18. Last year MIT math majors produced a tee shirt that said ${i\choose 18}$ ("I choose 8") on the front, and on the back $$ \frac{34376687+1499084559i}{14485008384}. $$ With the more natural denominator $18!$ this is $$ \frac{15194495654000+662595375078000i}{18!}. $$ This suggests the question: for any $n\geq 1$ find a "nice" combinatorial interpretation of the real and imaginary parts of $i(i-1)(i-2)\cdots (i-n+1)=f_n+ig_n$. It is easy to express $f_n$ and $g_n$ as certain alternating sums of Stirling numbers of the first kind, but I don't consider this "nice." The $g_n$'s seem to alternate in sign beginning with $n=5$. The $f_n$'s alternate in sign up to $n=17$ and then seem to alternate in sign beginning with $n=18$. It is curious that $i(i-1)(i-2)(i-3)=-10$, a real number. One could ask the same question with $i$ replaced by any Gaussian integer $a+bi$. One can also ask about the asymptotic rate of growth of $f_n$ and $g_n$. Clearly $f_n^2+g_n^2\sim C\cdot (n-1)!^2$, so one would expect $f_n$ and $g_n$ to be roughly of the size of $(n-1)!$.
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