Hankel Determinant Calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences $f$ 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 $LU$-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 $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, 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 $2^n$ in the infinite product defining the $\pm 1$ Thue--Morse sequence are replaced by $3^n$

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 tt-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 $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, 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 $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$

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