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Register a new userA bijective proof of the Cauchy identity for Grothendieck polynomials
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Authors
Yasuhide Numata
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We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and dual Grothendieck polynomials.
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We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $beta$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $n$ pairs of variables is a Drinfeld twist of the $U_q(widehat{mathfrak{sl}}_{n+1})$ $R$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $beta$-Grothendieck polynomials and dual double $beta$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillovs Cauchy identity for $beta$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $beta$-Grothendieck polynomials, and prove a new branching rule for double $beta$-Grothendieck polynomials.
Babson and Steingr{i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $pi$, let $des(pi)$ denote the descent number of $pi$ and $maj(pi)$ denote the major index of $pi$. Babson and Steingr{i}msson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.
We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of modern enumerative combinatorics and their applications to computational problems.
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s-1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s, s+2)$-core partitions with distinct parts and a set of lattice paths.
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