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Séminaire Lotharingien de Combinatoire, B63d (2010), 8 pp.

# Sivaramakrishnan Sivasubramanian

# Hankel Determinants of Some Sequences of Polynomials

**Abstract.**
Ehrenborg gave a combinatorial proof of Radoux's theorem which states
that the determinant of the
(*n*+1)x(*n*+1) dimensional Hankel matrix of exponential polynomials
is *x*^{n(n+1)/2} *\prod_{i=0}^n i*!.
This proof also shows
the result that the (*n*+1)x(*n*+1) Hankel matrix of factorial
numbers is *\prod_{k=1}^n* (*k*!)^{2}.
We observe that two polynomial
generalizations of factorial numbers
also have interesting determinant values for Hankel matrices.
A polynomial generalization of the determinant of the Hankel
matrix with entries being fixed-point free involutions on the set [2*n*]
is given next. We also give a bivariate non-crossing analogue of a theorem
of Cigler about the determinant of a similar Hankel matrix.

Received: October 4, 2009.
Revised: April 4, 2010.
Accepted: April 17, 2010.
Final Version: April 20, 2010.

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