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Let $P$ be the poset of all subsets of $\{1,2,\ldots, n\}$ with av even number of elements, ordered by inclusion. There is a recursive formula for the Möbius function on a poset: $$ \mu(x,y) = -\sum_{x\leq z < y} \mu(x,z) ~~~~ \textrm{when} ~x<y$$ and $$\mu(x,x)=1$$ Is it possible to find a explicit formula in the case of this particular poset? For the Boolean lattice (all subsets of $\{1,2,\ldots, n\}$) it is $\mu(X,Y)=(-1)^{|Y\setminus X|}$, but the property of the subsets having an even number of elements makes it more complicated.

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  • $\begingroup$ Experimentally, $\mu(X,Y) = a\big(\frac12\#(Y\setminus X)\big)$, where $a(n)$ are the signed Euler/secant numbers; see oeis.org/A028296 for more information that will presumably lead to a proof. $\endgroup$ – Greg Martin Nov 16 '14 at 23:01
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It is easy to see that $\mu(S,T)$ depends only on $\lvert T \setminus S \rvert$. Hence, $\mu(S,T) = \hat{\mu}(\lvert T \setminus S \rvert)$ for some function $\hat{\mu}$ defined on the non-negative even integers. We will show that \begin{equation*} \sum_{\substack{k=0\\\text{$k$ even}}}^{+\infty} \hat{\mu}(k) \frac{t^k}{k!} = \frac{1}{\cosh(t)} \:. \end{equation*} Indeed, \begin{equation*} \cosh(t) \sum_{\substack{k=0\\\text{$k$ even}}}^{+\infty} \hat{\mu}(k) \frac{t^k}{k!} = \sum_{\substack{j=0\\\text{$j$ even}}}^{+\infty} \frac{t^j}{j!} \sum_{\substack{k=0\\\text{$k$ even}}}^{+\infty} \hat{\mu}(k) \frac{t^k}{k!} = \sum_{\substack{j=0\\\text{$j$ even}}}^{+\infty} \sum_{\substack{k=0\\\text{$k$ even}}}^{+\infty} \hat{\mu}(k) \frac{t^{j+k}}{j!k!} \:. \end{equation*} And thus \begin{equation*} \cosh(t) \sum_{\substack{k=0\\\text{$k$ even}}}^{+\infty} \hat{\mu}(k) \frac{t^k}{k!} = \sum_{\substack{r=0\\\text{$r$ even}}}^{+\infty} \sum_{\substack{k=0\\\text{$k$ even}}}^{r} \hat{\mu}(k) \frac{t^{r}}{k!(r-k)!} = \sum_{\substack{r=0\\\text{$r$ even}}}^{+\infty} \frac{t^r}{r!}\sum_{\substack{k=0\\\text{$k$ even}}}^{r} \! \binom{r}{k} \: \hat{\mu}(k) = 1 \:. \end{equation*} We conclude with the Taylor expansion of $\frac{1}{\cosh(t)}$ involving the Euler's numbers.

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