Ribbon tile invariants from the signed area (Q1601417)

Ribbon tiles are polyominoes consisting of \(n\) squares of the square grid laid out along a path which, at each step, moves north or east. A ribbon tile invariant is a translation-invariant mapping \(\varphi\) on the family of ribbon tiles such that, if \(\Gamma\) is a simply-connected polyomino and \(\nu\) a tiling of \(\Gamma\) by ribbon tiles \(\tau\), then \(\sum_{\tau\in \nu} \varphi(\tau)\) depends only on \(\Gamma\) but not on \(\nu\). The authors establish the existence of a large class of ribbon tile invariants. In particular, they prove a conjecture about a full basis of ribbon tile invariants, made in \textit{I. Pak} [Trans. Am. Math. Soc. 352, 5525-5561 (2000; Zbl 0963.05030)]. Their results are applied to determine the structure of the tile counting group for ribbon tiles introduced in Pak's paper mentioned above.