We contribute new bi-symmetric equidistributions to this subject. Our main result is
a bijective proof of a bi-symmetric septuple equidistribution of
statistics on ascent sequences, involving
the number of ascents (`asc`),
the number of repeated entries (`rep`),
the number of zeros (`zero`),
the number of maximal entries (`max`),
the number of right-to-left minima (`rmin`),
and two additional statistics.
We further establish a new transformation formula for non-terminating
basic hypergeometric _{4}φ_{3} series expanded as an analytic function
in base *q* around *q*=1, which is utilized to prove two (bi)-symmetric
quadruple equidistributions on ascent sequences.
A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler-Stirling statistics
(`asc`,`rep`,`zero`,`max`) and
(`rep`,`asc`,`max`,`zero`) on ascent
sequences, that was motivated by a double Eulerian equidistribution due
to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the
first author (2018).

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

The following versions are available:

- PDF ( K)
- TeX version