30 research outputs found
The number of terms in the permanent and the determinant of a generic circulant matrix
Let A=(a_(ij)) be the generic n by n circulant matrix given by
a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp.
p(n)) to be the number of terms in the determinant (resp. permanent) of A. The
function p(n) is well-known and has several combinatorial interpretations. The
function d(n), on the other hand, has not been studied previously. We show that
when n is a prime power, d(n)=p(n). The proof uses symmetric functions.Comment: 6 pages; 1 figur
Descent c-Wilf Equivalence
Let denote the symmetric group. For any , we let
denote the number of descents of ,
denote the number of inversions of , and
denote the number of left-to-right minima of .
For any sequence of statistics on
permutations, we say two permutations and in are
-c-Wilf equivalent if the generating
function of over all permutations which
have no consecutive occurrences of equals the generating function of
over all permutations which have no
consecutive occurrences of . We give many examples of pairs of
permutations and in which are -c-Wilf
equivalent, -c-Wilf equivalent, and
-c-Wilf equivalent. For example, we
will show that if and are minimally overlapping permutations
in which start with 1 and end with the same element and
and , then and are
-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431
Irreducible Representations From Group Actions on Trees
We study the representations of the symmetric group found by acting on
labeled graphs and trees with vertices. Our main results provide
combinatorial interpretations that give the number of times the irreducible
representations associated with the integer partitions and appear
in the representations. We describe a new sign
reversing involution with fixed points that provide a combinatorial
interpretation for the number of times the irreducible associated with the
integer partition appears in the representations
Symmetric nonnegative forms and sums of squares
We study symmetric nonnegative forms and their relationship with symmetric
sums of squares. For a fixed number of variables and degree , symmetric
nonnegative forms and symmetric sums of squares form closed, convex cones in
the vector space of -variate symmetric forms of degree . Using
representation theory of the symmetric group we characterize both cones in a
uniform way. Further, we investigate the asymptotic behavior when the degree
is fixed and the number of variables grows. Here, we show that, in
sharp contrast to the general case, the difference between symmetric
nonnegative forms and sums of squares does not grow arbitrarily large for any
fixed degree . We consider the case of symmetric quartic forms in more
detail and give a complete characterization of quartic symmetric sums of
squares. Furthermore, we show that in degree the cones of nonnegative
symmetric forms and symmetric sums of squares approach the same limit, thus
these two cones asymptotically become closer as the number of variables grows.
We conjecture that this is true in arbitrary degree .Comment: (v4) minor revision and small reorganizatio
The combinatorics of Jeff Remmel
We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics