4 research outputs found
Fingerprint databases for theorems
We discuss the advantages of searchable, collaborative, language-independent
databases of mathematical results, indexed by "fingerprints" of small and
canonical data. Our motivating example is Neil Sloane's massively influential
On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater
mathematical community to search for the appropriate fingerprints within each
discipline, and to compile fingerprint databases of results wherever possible.
The benefits of these databases are broad - advancing the state of knowledge,
enhancing experimental mathematics, enabling researchers to discover unexpected
connections between areas, and even improving the refereeing process for
journal publication.Comment: to appear in Notices of the AM
Pinnacle sets of signed permutations
Pinnacle sets record the values of the local maxima for a given family of
permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual
concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent
years pinnacles and admissible pinnacles sets for the type symmetric group
have been widely studied. In this article we define the pinnacle set of signed
permutations of types and . We give a closed formula for the number of
type / admissible pinnacle sets and answer several other related
enumerative questions.Comment: 15 pages, 3 figures, to appear in Discrete Mathematic
RELATIONS ON GENERALIZED DEGREE SEQUENCES
We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley