1,899 research outputs found
Equivalence classes for the mu-coefficient of Kazhdan-Lusztig polynomials in S_n
We study equivalence classes relating to the Kazhdan-Lusztig mu(x,w)
coefficients in order to help explain the scarcity of distinct values. Each
class is conjectured to contain a "crosshatch" pair. We also compute the values
attained by mu(x,w) for the permutation groups S_10 and S_11.Comment: 13 pages, 6 figure
Juggling probabilities
The act of a person juggling can be viewed as a Markov process if we assume
that the juggler throws to random heights. I make this association for the
simplest reasonable model of random juggling and compute the steady state
probabilities in terms of the Stirling numbers of the second kind. I also
explore several alternate models of juggling.Comment: 11 pages, 5 eps figures. psfra
Gerrymandering and the net number of US House seats won due to vote-distribution asymmetries
Using the recently introduced declination function, we estimate the net
number of seats won in the US House of Representatives due to asymmetries in
vote distributions. Such asymmetries can arise from combinations of partisan
gerrymandering and inherent geographic advantage. Our estimates show
significant biases in favor of the Democrats prior to the mid 1990s and
significant biases in favor of Republicans since then. We find net differences
of 28, 20 and 25 seats in favor of the Republicans in the years 2012, 2014 and
2016, respectively. The validity of our results is supported by the technique
of simulated packing and cracking. We also use this technique to show that the
presidential-vote logistic regression model is insensitive to the packing and
cracking by which partisan gerrymanders are achieved.Comment: 18 pages, 9 figures. Revisions: Emphasized fact that S-declination
measures asymmetry in vote distribution and cannot account directly for
geographic clustering; added "greedy" packing/cracking algorithm; added
paragraph on simulations; other minor edits and correction
Optimized random chemistry
The random chemistry algorithm of Kauffman can be used to determine an
unknown subset S of a fixed set V. The algorithm proceeds by zeroing in on S
through a succession of nested subsets V=V_0,V_1,...,V_m=S. In Kauffman's
original algorithm, the size of each V_i is chosen to be half the size of
V_{i-1}. In this paper we determine the optimal sequence of sizes so as to
minimize the expected run time of the algorithm.Comment: 7 pages, 3 figures; added one reference, minor typos fixe
Counterexamples to the 0-1 conjecture
For permutations x and w, let mu(x,w) be the coefficient of highest possible
degree in the Kazhdan-Lusztig polynomial P_{x,w}. It is well-known that the
coefficients mu(x,w) arise as the edge labels of certain graphs encoding the
representations of S_n. The 0-1 Conjecture states that the mu(x,w) are either 0
or 1. We present two counterexamples to this conjecture, the first in S_16, for
which x and w are in the same left cell, and the second in S_10. The proof of
the counterexample in S_16 relies on computer calculations.Comment: 15 pages, 4 figures; code for computer calculations included in
source packag
Maximal singular loci of Schubert varieties in SL(n)/B
We give an explicit combinatorial description of the irreducible components
of the singular locus of the Schubert variety X_w for any element w in S_n. Our
description of the irreducible components is computationally more efficient
(O(n^6)) than the previously best known algorithms. This result proves a
conjecture of Lakshmibai and Sandhya regarding this singular locus.
Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig
polynomials at the maximum singular points.Comment: 50 pages, 50 figure
Nested quantum Dyck paths and nabla(s_lambda)
We conjecture a combinatorial formula for the monomial expansion of the image
of any Schur function under the Bergeron-Garsia nabla operator. The formula
involves nested labeled Dyck paths weighted by area and a suitable "diagonal
inversion" statistic. Our model includes as special cases many previous
conjectures connecting the nabla operator to quantum lattice paths. The
combinatorics of the inverse Kostka matrix leads to an elementary proof of our
proposed formula when q=1. We also outline a possible approach for proving all
the extant nabla conjectures that reduces everything to the construction of
sign-reversing involutions on explicit collections of signed, weighted objects.Comment: 23 page
Accumulation charts for instant-runoff elections
We propose a new graphical format for instant-runoff voting election results.
We call this proposal an "accumulation chart." This model, a modification of
standard bar charts, is easy to understand, clearly indicates the winner,
depicts the instant-runoff algorithm, and summarizes the votes cast. Moreover,
it includes the pedigree of each accumulated vote and gives a clear depiction
of candidates' coalitions.Comment: to appear in Notices of the AM
Matching expectations
The game of memory is played with a deck of n pairs of cards. The cards in
each pair are identical. The deck is shuffled and the cards laid face down. A
move consists of flipping over first one card then another. The cards are
removed from play if they match. Otherwise, they are flipped back over and the
next move commences. A game ends when all pairs have been matched. We determine
that, when the game is played optimally, as n tends to infinity: 1) The
expected number of moves is (3 - 2 ln 2)n + 7/8 - 2 ln 2 (approximately 1.61
n), 2) The expected number of times two matching cards are unwittingly flipped
over is ln 2, and 3) The expected number of flips until two matching cards have
been seen is asymptotically sqrt{pi n}.Comment: 16 page
Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations
We give a combinatorial formula for the Kazhdan-Lusztig polynomials
in the symmetric group when is a 321-hexagon-avoiding permutation. Our
formula, which depends on a combinatorial framework developed by Deodhar, can
be expressed in terms of a simple statistic on all subexpressions of any fixed
reduced expression for . We also show that being 321-hexagon-avoiding is
equivalent to several other conditions, such as the Bott-Samelson resolution of
the Schubert variety being small. We conclude with a simple method for
completely determining the singular locus of when is
321-hexagon-avoiding.Comment: 24 pages, 18 figures, AMS-LaTe
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