52,414 research outputs found
Euler-Mahonian triple set-valued statistics on permutations
The inversion number and the major index are equidistributed on the symmetric
group. This is a classical result, first proved by MacMahon, then by Foata by
means of a combinatorial bijection. Ever since many refinements have been
derived, which consist of adding new statistics, or replacing integral-valued
statistics by set-valued ones. See the works by Foata-Schutzenberger, Skandera,
Foata-Han and more recently by Hivert-Novelli-Thibon.
In the present paper we derive a general equidistribution property on
Euler-Mahonian set-valued statistics on permutations, which unifies the above
four refinements. We also state and prove the so-called "complement property"
of the Majcode
Hankel Determinant Calculus for the Thue-Morse and related sequences
The Hankel determinants of certain automatic sequences are evaluated,
based on a calculation modulo a prime number. In most cases, the Hankel
determinants of automatic sequences do not have any closed-form expressions;
the traditional methods, such as -decompo\-si\-tion and Jacobi continued
fraction, cannot be applied directly. Our method is based on a simple idea: the
Hankel determinants of each sequence equal to modulo are equal to
the Hankel determinants of modulo . The clue then consists of finding a
nice sequence , whose Hankel determinants have closed-form expressions.
Several examples are presented, including a result saying that the Hankel
determinants of the Thue-Morse sequence are nonzero, first proved by Allouche,
Peyri\`ere, Wen and Wen using determinant manipulation. The present approach
shortens the proof of the latter result significantly. We also prove that the
corresponding Hankel determinants do not vanish when the powers in the
infinite product defining the Thue--Morse sequence are replaced by
From charge transfer type insulator to superconductor
We propose a microscopic model Hamiltonian to account for impurity doping
induced insulator-superconductor transition and the coexistence of
antiferromagnetism and superconductivity in the high-Tc cuprates. The crossover
from non Fermi liquid to Fermi liquid regime characterized by delocalization of
delocalization of d electrons at Cu sites is discussed
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