The gfeqns function returns a list of the generating function equations which count the objects described in the specification.

Each nonterminal has an associated equation which uses the nonterminal name as the name of the equation. For example, $A\left(z\right)$ would be the generating function for $A$. The equation is in terms of the other nonterminals.

If the objects are labeled, the exponential generating functions are produced. If the objects are unlabeled, ordinary generating functions are used.

Objects can be marked (tagged) by forming the product of the object with a named Epsilon and associating a tag with that Epsilon.

If the tag is a variable name, the resulting generating function equations have an extra variable that marks that object. The same tag can be associated with more than one Epsilon name. The tag does not need to be a variable.

For information on how to write specifications, see combstruct and combstruct[specification]

$\mathrm{with}\left(\mathrm{combstruct}\right)\:$

$\mathrm{tree}≔\left\{L=\mathrm{Atom}\,N=\mathrm{Atom}\,T=\mathrm{Union}\left(L\,\mathrm{Prod}\left(N\,T\,T\right)\right)\right\}\:$

$\mathrm{gfeqns}\left(\mathrm{tree}\,\mathrm{labeled}\,z\right)$

$\left[L\left(z\right)=z\,N\left(z\right)=z\,T\left(z\right)=z+z{T\left(z\right)}^2\right]$

$\mathrm{tree1}≔\left\{L=\mathrm{Prod}\left(\mathrm{leaf}\,\mathrm{Atom}\right)\,N=\mathrm{Atom}\,T=\mathrm{Union}\left(L\,\mathrm{Prod}\left(N\,\mathrm{Set}\left(T\right)\right)\right)\,\mathrm{leaf}=\mathrm{{\rm E}}\right\}\:$

$\mathrm{gfeqns}\left(\mathrm{tree1}\,\mathrm{unlabeled}\,z\,\left[\left[u\,\mathrm{leaf}\right]\right]\right)$

$\left[L\left(z\,u\right)=uz\,N\left(z\,u\right)=z\,T\left(z\,u\right)=L\left(z\,u\right)+z{\ⅇ}^{\sum _{j_1=1}^{\mathrm{\infty }}\frac{T\left(z^{j_1}\,u^{j_1}\right)}{j_1}}\,\mathrm{leaf}\left(z\,u\right)=u\right]$

$\mathrm{gfeqns}\left(\mathrm{tree1}\,\mathrm{labeled}\,z\,\left[\left[2\,\mathrm{leaf}\right]\right]\right)$

$\left[L\left(z\right)=2z\,N\left(z\right)=z\,T\left(z\right)=L\left(z\right)+z{\ⅇ}^{T\left(z\right)}\,\mathrm{leaf}\left(z\right)=2\right]$