On the sign-imbalance of partition shapes

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.Comment: 12 pages. Better presentatio

Making multigraphs simple by a sequence of double edge swaps

We show that any loopy multigraph with a graphical degree sequence can be transformed into a simple graph by a finite sequence of double edge swaps with each swap involving at least one loop or multiple edge. Our result answers a question of Janson motivated by random graph theory, and it adds to the rich literature on reachability of double edge swaps with applications in Markov chain Monte Carlo sampling from the uniform distribution of graphs with prescribed degrees.Comment: 13 page

Learning to Personalize in Appearance-Based Gaze Tracking

Personal variations severely limit the performance of appearance-based gaze tracking. Adapting to these variations using standard neural network model adaptation methods is difficult. The problems range from overfitting, due to small amounts of training data, to underfitting, due to restrictive model architectures. We tackle these problems by introducing the SPatial Adaptive GaZe Estimator (SPAZE). By modeling personal variations as a low-dimensional latent parameter space, SPAZE provides just enough adaptability to capture the range of personal variations without being prone to overfitting. Calibrating SPAZE for a new person reduces to solving a small optimization problem. SPAZE achieves an error of 2.70 degrees with 9 calibration samples on MPIIGaze, improving on the state-of-the-art by 14 %. We contribute to gaze tracking research by empirically showing that personal variations are well-modeled as a 3-dimensional latent parameter space for each eye. We show that this low-dimensionality is expected by examining model-based approaches to gaze tracking. We also show that accurate head pose-free gaze tracking is possible

Correlation Effects in Orbital Magnetism

Orbital magnetization is known empirically to play an important role in several magnetic phenomena, such as permanent magnetism and ferromagnetic superconductivity. Within the recently developed ''modern theory of orbital magnetization'', theoretical insight has been gained into the nature of this often neglected contribution to magnetism, but is based on an underlying mean-field approximation. From this theory, a few treatments have emerged which also take into account correlations beyond the mean-field approximation. Here, we apply the scheme developed in a previous work [Phys. Rev. B ${\bf \text{93}}$, 161104(R) (2016)] to the Haldane-Hubbard model to investigate the effect of charge fluctuations on the orbital magnetization within the $GW$ approximation. Qualitatively, we are led to distinguish between two quite different situations: (i) When the lattice potential is larger than the nearest neighbor hopping, the correlations are found to boost the orbital magnetization. (ii) If the nearest neighbor hopping is instead larger than the lattice potential, the correlations reduce the magnetization.Comment: 8 pages, 9 figure

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w\in \Sn is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.Comment: 24 page