From Fibonacci to Robbins: Series Reversion and Hankel Transforms
Paul Barry
School of Science
Waterford Institute of Technology
Ireland
Abstract:
The Robbins numbers An
have an important place in the study of plane
partitions and in the study of alternating sign matrices. A simple
closed formula exists for these numbers, but its derivation entails
quite sophisticated machinery. Building on this basis, we study the
Robbins numbers from a more elementary standpoint, based on series
reversion and Hankel transforms. We show how a transformation pipeline
can lead from the Fibonacci numbers to the Robbins numbers. We employ
the language of Riordan arrays to carry out many of the transformations
of the generating functions that we use. We establish links between the
revert transforms under discussion and certain scaled moment sequences of
a family of continuous Hahn polynomials. Finally we show that a family
of quasi-Fibonacci polynomials of 7th order play a fundamental role in
this theory.
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(Concerned with sequences
A000045
A000108
A000364
A005130
A005156
A007318
A051255
A052547
A052975
A055209
A077998
A121449
A292865.)
Received April 4 2021; revised version received November 17 2021.
Published in Journal of Integer Sequences,
November 19 2021.
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