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We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $genfrac{[}{]}{0pt}{}{n}{k}_{q}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our $q$-$(1+q)$-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birkhoff transform of any finite poset. We end with a discussion of higher analogues of the $q$-binomial.
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Tail dependence refers to clustering of extreme events. In the context of financial risk management, the clustering of high-severity risks has a devastating effect on the well-being of firms and is thus of pivotal importance in risk analysis.When it comes to quantifying the extent of tail dependence, it is generally agreed that measures of tail dependence must be independent of the marginal distributions of the risks but rather solely copula-dependent. Indeed, all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copulas domain of definition.In this paper we urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. For the Gaussian copula, however, we prove that the classical measures are maximal. The implication of the result is two-fold: On the one hand, it means that in the Gaussian case, the (weak) measures of tail dependence that have been reported and used are of utmost prudence, which must be a reassuring news for practitioners. On the other hand, it further encourages substitution of the Gaussian copula with other copulas that are more tail dependent.
It is well known that for all $ngeq1$ the number $n+ 1$ is a divisor of the central binomial coefficient ${2nchoose n}$. Since the $n$th central binomial coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. The Chung-Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for $2n-1$, another divisor of ${2nchoose n}$. We then show our main result follows from a more general observation regarding binomial coefficients ${nchoose k}$ with $n$ and $k$ relatively prime. A discussion of the case where $n$ and $k$ are not relatively prime is also given, highlighting the limitations of our methods. Finally, we come full circle and give a novel interpretation of the Catalan numbers.
In this note we show how to rederive the $A_2$ Rogers-Ramanujan identities proven by Andrews, Schilling and Warnaar using cylindric partitions. This paper is dedicated to George Andrews for his $80^{th}$ birthday.
Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution-operator displays fluctuations around itsexpectation. The recently-developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $dge 1$ previous results so far limited to dimension $d=1$.
We give combinatorial proofs of $q$-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitzs identity, a new proof of the $q$-Frobenius identity of Garsia and Remmel and of Ehrenborgs Hankel $q$-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.
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