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 $P_{x,w}$ in the symmetric group when $w$ 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 $w$. We also show that $w$ being 321-hexagon-avoiding is equivalent to several other conditions, such as the Bott-Samelson resolution of the Schubert variety $X_w$ being small. We conclude with a simple method for completely determining the singular locus of $X_w$ when $w$ is 321-hexagon-avoiding.Comment: 24 pages, 18 figures, AMS-LaTe