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The k-orbit reconstruction and the orbit algebra

Abstract

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 ≤ iN, there exists k, 1 ≤ kn, 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).

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Mnukhin, V.B. The k-orbit reconstruction and the orbit algebra. Acta Appl Math 29, 83–117 (1992). https://doi.org/10.1007/BF00053380

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Mathematics Subject Classifications (1991)

  • 05C60
  • 05E20
  • 20B25
  • 20B99
  • 94B27