5,903 research outputs found
The Andrews-Gordon identities and -multinomial coefficients
We prove polynomial boson-fermion identities for the generating function of
the number of partitions of of the form , with
, and . The bosonic side of
the identities involves -deformations of the coefficients of in the
expansion of . A combinatorial interpretation for these
-multinomial coefficients is given using Durfee dissection partitions. The
fermionic side of the polynomial identities arises as the partition function of
a one-dimensional lattice-gas of fermionic particles. In the limit
, our identities reproduce the analytic form of Gordon's
generalization of the Rogers--Ramanujan identities, as found by Andrews. Using
the duality, identities are obtained for branching functions
corresponding to cosets of type of fractional level .Comment: 31 pages, Latex, 9 Postscript figure
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Canonical characters on quasi-symmetric functions and bivariate Catalan numbers
Every character on a graded connected Hopf algebra decomposes uniquely as a
product of an even character and an odd character (Aguiar, Bergeron, and
Sottile, math.CO/0310016).
We obtain explicit formulas for the even and odd parts of the universal
character on the Hopf algebra of quasi-symmetric functions. They can be
described in terms of Legendre's beta function evaluated at half-integers, or
in terms of bivariate Catalan numbers:
Properties of characters and of quasi-symmetric functions are then used to
derive several interesting identities among bivariate Catalan numbers and in
particular among Catalan numbers and central binomial coefficients
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