Twisted product formulas for surgery with coefficients

1990 ◽  
pp. 115-140
Author(s):  
Karl Heinz Dovermann ◽  
Reinhard Schultz
Keyword(s):  
Twisted Product ◽  
Product Formulas

scholarly journals On doubly twisted product immersions

2021 ◽  
Author(s):  
Abdoul Salam Diallo ◽  
Fortuné Massamba
Keyword(s):  
Twisted Product

scholarly journals Augmented Rook Boards and General Product Formulas

10.37236/809 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Brian K. Miceli ◽  
Jeffrey Remmel
Keyword(s):  
Product Formula ◽  
Combinatorial Proof ◽  
Rook Theory ◽  
Special Cases ◽  
Ferrers Board ◽  
General Product ◽  
Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

scholarly journals Brick Manifolds and Toric Varieties of Brick Polytopes

10.37236/5038 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Moment Map ◽  
Twisted Product ◽  
General Fiber ◽  
Moment Polytope ◽  
The Moment ◽  
Subword Complex ◽  
Do So

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

scholarly journals Evaluating the Numbers of some Skew Standard Young Tableaux of Truncated Shapes

10.37236/3890 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Young Tableaux ◽  
Multiple Integrals ◽  
Standard Young Tableaux ◽  
Product Formulas

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT $((n+k)^{r+1},n^{m-1}) / (n-1)^r $ truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape $((n+1)^3,n^{m-2}) / (n-2) \backslash \; (2^2)$, and the SYT of truncated shape $((n+1)^2,n^{m-2}) \backslash \; (2)$.

Semi-Twisted Product of Constant Curvature

2005 ◽  
Vol 277-279 ◽  
pp. 753-756
Author(s):  
Mi Suk Kim ◽  
Mi Hye Kim ◽  
Yong Ho Yon
Keyword(s):  
Constant Curvature ◽  
Twisted Product

Evolution of the Robertson–Walker metric under 2-loop renormalization group flow

2017 ◽  
Vol 26 (03) ◽  
pp. 1750021
Author(s):  
F. Hesamifard ◽  
M. M. Rezaii
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Ricci Flow ◽  
Necessary Condition ◽  
Renormalization Group Flow ◽  
Twisted Product ◽  
Walker Metric ◽  
Group Flow

Here, we study the evolution of a Robertson–Walker (RW) metric under the Ricci flow and 2-loop renormalization group flow (RG-2 flow). We show that a RW metric is a fixed point of the Ricci flow and it is not a solution of the RG-2 flow. RG-2 flow is considered on a doubly twisted product metric with further assumptions and also we introduce a necessary condition for existence of the solution of RG-2 flow.

Weakly Berwald Doubly-Twisted Product Finsler Metrics

2021 ◽  
Vol 11 (07) ◽  
pp. 1389-1399
Author(s):  
香香 邓
Keyword(s):  
Finsler Metrics ◽  
Twisted Product
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