MacMahon's theorem for a set of permutations with given descent indices and right-maximal records.
q-Leibniz Algebras
2000 Mathematics Subject Classification: Primary 17A32, Secondary 17D25. An algebra (A,ο) is called Leibniz if aο(bοc) = (a ο b)ο c-(a ο c) ο b for all a,b,c ∈ A. We study identities for the algebras A(q) = (A,οq), where a οq b = a ο b+q b ο a is the q-commutator. Let Char K ≠ 2,3. We show that the class of q-Leibniz algebras is defined by one identity of degree 3 if q2 ≠ 1, q ≠−2, by two identities of degree 3 if q = −2, and by the commutativity identity and one identity of degree 4...
Free Bicommutative Algebras
2000 Mathematics Subject Classification: Primary 17A50, Secondary 16R10, 17A30, 17D25, 17C50. Algebras with identities a(bc)=b(ac), (ab)c=(ac)b is called bicommutative. Bases and the cocharacter sequence for free bicommutative algebras are found. It is shown that the exponent of the variety of bicommutaive algebras is equal to 2.
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