Related papers: $r-$Bell polynomials in combinatorial Hopf algebra…
We extend the previously established zesting techniques from fusion categories to general tensor categories. In particular we consider the category of comodules over a Hopf algebra, providing a detailed translation of the categorical…
The aim of this paper is to study some basic properties of the r$-extended Lah-Bell numbers and polynomials associated with r-Lah numbers and to show the connection between the r-extended Lah-Bell polynomials and the rising factorial…
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…
We give some examples of, and raise some questions on, extensions of semisimple Hopf algebras.
We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational…
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We…
The nth r-extended Lah-Bell number is defined as the number of ways a set with $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to…
In this paper we revisit the work of E.T. Bell concerning partition polynomials in order to introduce the reciprocal partition polynomials. We give their explicit formulas and apply the result to compute closed formulae for some well-known…
We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations…
We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter…
In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six…
The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the…
In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there…
The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings…
New types of bialgebras arising from the Hopf equation (pentagonal equation) are introduced and studied. They will play from the Hopf equation the same role as the co-quasitriangular do from the quantum Yang Baxter equation.
We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of Grinberg's results…
We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and…
We introduce a Combinatorial Hopf Algebra (CHA) with bases indexed by the partition diagrams indexing the bases for partition algebras. By analogy with the operation $H_{\alpha} H_{\beta} = H_{\alpha \cdot \beta}$ for the complete…
We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a…
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the…