12,407 research outputs found
A q-Analog of Dual Sequences with Applications
In the present paper combinatorial identities involving q-dual sequences or
polynomials with coefficients q-dual sequences are derived. Further,
combinatorial identities for q-binomial coefficients(Gaussian coefficients),
q-Stirling numbers and q-Bernoulli numbers and polynomials are deduced.Comment: 14 page
Comultiplication rules for the double Schur functions and Cauchy identities
The double Schur functions form a distinguished basis of the ring
\Lambda(x||a) which is a multiparameter generalization of the ring of symmetric
functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended
to \Lambda(x||a) in a natural way so that the double power sums symmetric
functions are primitive elements. We calculate the dual Littlewood-Richardson
coefficients in two different ways thus providing comultiplication rules for
the double Schur functions. We also prove multiparameter analogues of the
Cauchy identity. A new family of Schur type functions plays the role of a dual
object in the identities. We describe some properties of these dual Schur
functions including a combinatorial presentation and an expansion formula in
terms of the ordinary Schur functions. The dual Littlewood-Richardson
coefficients provide a multiplication rule for the dual Schur functions.Comment: 44 pages, some corrections are made in sections 2.3 and 5.
Bell polynomials in combinatorial Hopf algebras
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934.
These polynomials have numerous applications in Combinatorics, Analysis,
Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve
combinatorial objects (set partitions, set partitions in lists, permutations,
etc.). So it seems natural to investigate analogous formulae in some
combinatorial Hopf algebras with bases indexed by these objects. The algebra of
symmetric functions is the most famous example of a combinatorial Hopf algebra.
In a first time, we show that most of the results on Bell polynomials can be
written in terms of symmetric functions and transformations of alphabets. Then,
we show that these results are clearer when stated in other Hopf algebras (this
means that the combinatorial objects appear explicitly in the formulae). We
investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the
Lagrange-B{\"u}rmann formula
The Semigroups B\u3csub\u3e2\u3c/sub\u3e and B\u3csub\u3e0\u3c/sub\u3e are Inherently Nonfinitely Based, as Restriction Semigroups
The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, forgetting the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable modulo monoids .
These results are consequences of — and discovered as a result of — an analysis of varieties of strict restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of completely r-semisimple restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation �. For example, explicit bases of identities are found for the varieties generated by B0 and B2
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
Multiplicate inverse forms of terminating hypergeometric series
The multiplicate form of Gould--Hsu's inverse series relations enables to
investigate the dual relations of the Chu-Vandermonde-Gau{\ss}'s, the
Pfaff-Saalsch\"utz's summation theorems and the binomial convolution formula
due to Hagen and Rothe. Several identitity and reciprocal relations are thus
established for terminating hypergeometric series. By virtue of the duplicate
inversions, we establish several dual formulae of Chu-Vandermonde-Gau{\ss}'s
and Pfaff-Saalsch\"utz's summation theorems in Section (3)\cite{ChuVanGauss}
and (4)\cite{PfaffSaalsch}, respectively. Finally, the last section is devoted
to deriving several identities and reciprocal relations for terminating
balanced hypergeometric series from Hagen-Rothe's convolution identity in
accordance with the duplicate, triplicate and multiplicate inversions.Comment: 15 page
Unimodal Sequence Generating Functions Arising from Partition Ranks
In this paper we study generating functions resembling the rank of strongly
unimodal sequences. We give combinatorial interpretations, identities in terms
of mock modular forms, asymptotics, and a parity result. Our functions imitate
a relation between the rank of strongly unimodal sequences and the rank of
integer partitions
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