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We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials., 10 pages, 6 figures

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R -polynomials for the symmetric group. Let SnSn be the symmetric group on {1,2,…,n}{1,2,…,n}, and let S={si|1≤i≤n−1} be the generating set of SnSn, where for 1≤i≤n−11≤i≤n−1, sisi is the adjacent transposition. For a subset J⊆SJ⊆S, let (Sn)J(Sn)J be the parabolic subgroup generated by J , and let (Sn)J(Sn)J be the set of minimal coset representatives for Sn/(Sn)JSn/(Sn)J. For u≤v∈(Sn)Ju≤v∈(Sn)J in the Bruhat order and x∈{q,−1}x∈{q,−1}, let Ru,vJ,x(q) denote the parabolic R-polynomial indexed by u and v . Brenti found a formula for Ru,vJ,x(q) when J=S∖{si}J=S∖{si}, and obtained an expression for Ru,vJ,x(q) when J=S∖{si−1,si}J=S∖{si−1,si}. In this paper, we provide a formula for Ru,vJ,x(q), where J=S∖{si−2,si−1,si}J=S∖{si−2,si−1,si} and i appears after i−1i−1 in v. It should be noted that the condition that i appears after i−1i−1 in v is equivalent to that v is a permutation in (Sn)S∖{si−2,si}(Sn)S∖{si−2,si}. We also pose a conjecture for Ru,vJ,x(q), where J=S∖{sk,sk+1,…,si}J=S∖{sk,sk+1,…,si} with 1≤k≤i≤n−11≤k≤i≤n−1 and v is a permutation in (Sn)S∖{sk,si}(Sn)S∖{sk,si}.

The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition $\lambda$ and a positive integer $k$, let ${\mathrm{BSSYT}}(\lambda,k)$ (respectively, ${\mathrm{SYT}}(\lambda,k)$) denote the set of barely set-valued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape $\lambda$ with entries in row $i$ not exceeding $k+i$. In the case when $\lambda$ is a rectangular staircase partition $\delta_d(b^a)$, Reiner, Tenner and Yong conjectured that $|{\mathrm{BSSYT}}(\lambda,k)|= \frac{kab(d-1)}{(a+b)} |{\mathrm{SYT}}(\lambda,k)|$. In this paper, we establish a connection between barely set-valued tableaux and reverse plane partitions with designated corners. We show that for any shape $\lambda$, the expected jaggedness of a subshape of $\lambda$ under the weak probability distribution can be expressed as $\frac{2|{\mathrm{BSSYT}}(\lambda,k)|} {k|{\mathrm{SYT}}(\lambda,k)|}$. On the other hand, when $\lambda$ is a balanced shape with $r$ rows and $c$ columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in $\lambda$ under the weak distribution equals $2rc/(r+c)$. Hence, for a balanced shape $\lambda$ with $r$ rows and $c$ columns, we establish the relation that $|{\mathrm{BSSYT}}(\lambda,k)|=\frac{krc}{(r+c)}|{\mathrm{SYT}}(\lambda,k)|$. Since a rectangular staircase shape $\delta_d(b^a)$ is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong., Comment: 13 pages, 6 figures

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\)-polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.

We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation @p with cyclic major code (s"1,s"2,...,s"n) to a permutation @s with cyclic inversion code (s"1,s"2,...,s"n). We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Bjorner and Wachs.

Aiming at the range image obtained through multi-line structured light, a regional interpolation method is put forward in this paper. This method divides interpolation into two parts according to the memory format of the scattered data, one is interpolation of the data on the stripes, and the other is interpolation of data between the stripes. Trend interpolation method is applied to the data on the stripes, and Gauss wavelet interpolation method is applied to the data between the stripes. Experiments prove regional interpolation method feasible and practical, and it also promotes the speed and precision.

Motion estimation search occupies most time of H.264 encode, Full Integer Motion Search is adopted by motion estimation search in encode model. But the asthmatic quantity of Full Integer Motion Search is too large, and could not be applied in some real time system. Fast Integer Motion Search suitable for H.264 is realized in this paper. This algorithm utilizes different motion characteristics of different images, adopts different search algorithms according as images with different motion characteristic separately. The test results show that this algorithm reduces 84.92% of motion estimation time when the Signal-to-Noise decrease less than 0.1%.

The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by $\{s_{m,n}\}_{n\geq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$, where p is an integer satisfying $0\leq p\leq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$., 18 pages

The Springer numbers are defined in connection with the irreducible root systems of type $B_n$, which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of Andre signed permutations, and by Arnol'd in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length n and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number B(n,k) of labeled ballot paths starting at (0,0) and ending at (n,k). Using our bijection, we find a statistic $\alpha$ such that the number of snakes $\pi$ of type $B_n$ with $\alpha(\pi)=k$ equals B(n,k). We also show that our bijection specializes to a bijection between labeled Dyck paths of length 2n and alternating permutations on [2n]., Comment: 16 pages, 4 figures