The k-orbit reconstruction and the orbit algebra

Let (G, W) be a permutation group on a finite set W = {w ₁,..., w _n}. We consider the natural action of G on the set of all subsets of W. Let h ₀, h ₁,..., h _N be the orbits of this action. For each i, 1 i N, there exists k, 1 k n, such that h _i is a set of k-element subsets of W. In this case h _i is called a symmetrized k-orbit of the group (G, W) or simply a k-orbit. With a k-orbit h _i we associate a multiset H(h _i ) = % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeoaaa!3690!\\langle \]h _i ⁽¹⁾, h _i ⁽²⁾,..., h _i ^(k)% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJepaaa!36A1!\\rangle \] of its (k – 1)-suborbits. Orbits h _i and h _j are called equivalent if H(h _i ) = H(h _j ). An orbit is reconstructible if it is equivalent to itself only. The paper concerns the k-orbit reconstruction problem and its connections with different problems in combinatorics. The technique developed is based on the notion of orbit and co-orbit algebras associated with a given permutation group (G, W).