Manuscript Oai211-A@1 · Oai211-A.tex · definition · 3f3e4f63

Definition: Weighted corner-substitution contract

Conditional transfer to weighted planar-map blocks · lines 1228–1256

Oai211-A@1#def:blockcontract, Oai211-A@1#eq:substitution

A weighted rooted planar-map class supplies:

one trivial zero-edge map of weight one;
rooted nontrivial blocks, including the one-link and one-loop blocks, with positive weights $w (B)$ ;
a unique recursive corner substitution, with one child slot per block corner and trivial insertions retained as leaves;
multiplicative map weight equal to the product of its block weights;
a deterministic order of each rooted block's $2 m$ corners when it has $m$ edges.

Write

a_{n} = ∣ E (M) ∣ = n \sum w (M), b_{m} = ∣ E (B) ∣ = m \sum w (B), b_{0} = 1,

T (z) = n \geq 0 \sum a_{n} z^{n}, Φ (y) = 1 + m \geq 1 \sum b_{m} y^{m} .

Then the substitution contract gives the formal identity

T (z) = Φ (z T (z)^{2}) .

Displayed formulas in this passage

Oai211-A@1#eq:substitution · line 1252
$T (z) = Φ (z T (z)^{2}) .$
unlabeled · line 1243
$a_{n} = ∣ E (M) ∣ = n \sum w (M), b_{m} = ∣ E (B) ∣ = m \sum w (B), b_{0} = 1,$
unlabeled · line 1247
$T (z) = n \geq 0 \sum a_{n} z^{n}, Φ (y) = 1 + m \geq 1 \sum b_{m} y^{m} .$