12 research outputs found
A matroid-friendly basis for the quasisymmetric functions
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is
presented. It is shown to have nonnegative structure constants, and several
interesting properties relative to the space of quasisymmetric functions
associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and
Reiner. In particular, for loopless matroids, this basis reflects the grading
by matroid rank, as well as by the size of the ground set. It is shown that the
morphism F is injective on the set of rank two matroids, and that
decomposability of the quasisymmetric function of a rank two matroid mirrors
the decomposability of its base polytope. An affirmative answer is given to the
Hilbert basis question raised by Billera, Jia, and Reiner.Comment: 25 pages; exposition tightened, typos corrected; to appear in the
Journal of Combinatorial Theory, Series
Super quasi-symmetric functions via Young diagrams
We consider the multivariate generating series of -partitions in
infinitely many variables . For some family of ranked posets
, it is natural to consider an analog with two infinite alphabets.
When we collapse these two alphabets, we trivially recover . Our main
result is the converse, that is, the explicit construction of a map sending
back onto . We also give a noncommutative analog of the latter. An
application is the construction of a basis of WQSym with a non-negative
multiplication table, which lifts a basis of QSym introduced by K. Luoto.Comment: 12 pages, extended abstract of arXiv:1312.2727, presented at FPSAC
conference. The presentation of the results is quite different from the long
versio
Cyclic inclusion-exclusion and the kernel of P -partitions
International audienceFollowing the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs
Comparing skew Schur functions: a quasisymmetric perspective
Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A
and s_B are equal, then the skew shapes A and B must have the same "row overlap
partitions." Here we show that these row overlap equalities are also implied by
a much weaker condition than skew Schur equality: that s_A and s_B have the
same support when expanded in the fundamental quasisymmetric basis F.
Surprisingly, there is significant evidence supporting a conjecture that the
converse is also true.
In fact, we work in terms of inequalities, showing that if the F-support of
s_A contains that of s_B, then the row overlap partitions of A are dominated by
those of B, and again conjecture that the converse also holds. Our evidence in
favor of these conjectures includes their consistency with a complete
determination of all F-support containment relations for F-multiplicity-free
skew Schur functions. We conclude with a consideration of how some other
quasisymmetric bases fit into our framework.Comment: 26 pages, 7 figures. J. Combin., to appear. Version 2 includes a new
subsection (5.3) on a possible skew version of the Saturation Theore
Super quasi-symmetric functions via Young diagrams
We consider the multivariate generating series of partitions in infinitely many variables . For some family of ranked posets , it is natural to consider an analog with two infinite alphabets. When we collapse these two alphabets, we trivially recover . Our main result is the converse, that is, the explicit construction of a map sending back onto . We also give a noncommutative analog of the latter. An application is the construction of a basis of with a non-negative multiplication table, which lifts a basis of introduced by K. Luoto
Cyclic inclusion-exclusion
Following the lead of Stanley and Gessel, we consider a morphism which
associates to an acyclic directed graph (or a poset) a quasi-symmetric
function. The latter is naturally defined as multivariate generating series of
non-decreasing functions on the graph. We describe the kernel of this morphism,
using a simple combinatorial operation that we call cyclic inclusion-exclusion.
Our result also holds for the natural noncommutative analog and for the
commutative and noncommutative restrictions to bipartite graphs. An application
to the theory of Kerov character polynomials is given.Comment: comments welcom
Cyclic inclusion-exclusion and the kernel of P -partitions
Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs
Quasi-symmetric functions as polynomial functions on Young diagrams
We determine the most general form of a smooth function on Young diagrams,
that is, a polynomial in the interlacing or multirectangular coordinates whose
value depends only on the shape of the diagram. We prove that the algebra of
such functions is isomorphic to quasi-symmetric functions, and give a
noncommutative analog of this result.Comment: 34 pages, 4 figures, version including minor modifications suggested
by referee