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 4φ3 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
(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).
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