Alternate formula

For integers \(0 \le a \le b\) let \(E(a,b)\) be the number of magma equations \(t_1 = t_2\) with \(t_1\) of order \(a\), \(t_2\) of order \(b\), counting up to relabelling, up to switching terms, and only allowing the equation \(x=x\) of the form \(t=t\). Then one has

• \(E(0,0) = 2\).

• If \(a\neq b\):

  \[ E(a,b) = T(a) \cdot T(b) \cdot \sum_{\begin{smallmatrix} 1 \le p \le a+1 \\ 1 \le q \le b+1 \\ 0 \le s \le \min(p,q) \end{smallmatrix}} \left\{\begin{array}{c} a+1 \\ p \end{array}\right\} \left\{\begin{array}{c} b+1 \\ q \end{array}\right\} \binom{p}{s} \binom{q}{s} s! \ . \]

• If \(a= b > 0\):

  \[ \begin{aligned} E(a,b) &= \frac{1}{2} T(a)^2 \cdot \sum_{\begin{smallmatrix}1 \le p,q \le a+1\\ 0 \le s \le \min(p,q)\end{smallmatrix}} \left\{\begin{array}{c} a+1 \\ p \end{array}\right\} \left\{\begin{array}{c} a+1 \\ q \end{array}\right\} \binom{p}{s} \binom{q}{s} s! \\ &\quad + \frac{1}{2} T(a) \cdot \sum_{\begin{smallmatrix}1 \le p \le a+1\\ 0 \le s \le p\end{smallmatrix}} \left\{\begin{array}{c} a+1 \\ p \end{array}\right\} \binom{p}{s} \, {\sf Invol}(s) \\ &\quad - T(a) \cdot \sum_{1 \le p \le a+1} \left\{\begin{array}{c} a+1 \\ p \end{array}\right\}. \end{aligned} \]

Here the Stirling numbers \(\left\{\substack{n\\m}\right\}\) of the second kind count the number of ways to partition a set of \(n\) elements into \(m\) non-empty subsets,

\[ T(n) = \frac{1}{n+1} \cdot \binom{2n}{n} \]

counts the number of plane binary trees of order \(n\), and.

\[ {\sf Invol}(n) = \sum_{0 \leq k \leq \lfloor n/2 \rfloor} \binom{n}{2k} (2k-1)!! \]

counts the number of involutions on \(n\) elements.

The number of magma equations of order \(n\) (under the given constraints) is then equal to

\[ \sum_{0 \le a \le \lfloor n/2 \rfloor } E(a,n-a) . \]

For further details, as well as Maple calculations for \(E(a,b)\), see this folder.

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