3,950 research outputs found
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
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
On -extensions of the Hankel determinants of certain automatic sequences
In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse
sequence, and proved that all the Hankel determinants of the period-doubling
sequence are odd integral numbers. We speak of -extension when the entries
along the diagonal in the Hankel determinant are all multiplied by~. Then we
prove that the -extension of each Hankel determinant of the period-doubling
sequence is a polynomial in , whose leading coefficient is the {\it only
one} to be an odd integral number. Our proof makes use of the combinatorial
set-up developed by Bugeaud and Han, which appears to be very suitable for this
study, as the parameter counts the number of fixed points of a permutation.
Finally, we prove that all the -extensions of the Hankel determinants of the
regular paperfolding sequence are polynomials in of degree less than or
equal to
Specializations and Extensions of the quantum MacMahon Master Theorem
We study some specializations and extensions of the quantum version of the
MacMahon Master Theorem derived by Garoufalidis, Le and Zeilberger. In
particular, we obtain a (t,q)-analogue for the Cartier-Foata noncommutative
version and a semi-strong (t,q)-analogue for the contextual algebra.Comment: 12 page
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