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A MODEL FOR THE DOUBLE COSETS OF YOUNG SUBGROUPS
"... I am very grateful to Darij Grinberg for many corrections to errors in these note and helpful suggestions for clarifying remarks. Of course I have full responsibility for any remaining errors. 1. ..."
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I am very grateful to Darij Grinberg for many corrections to errors in these note and helpful suggestions for clarifying remarks. Of course I have full responsibility for any remaining errors. 1.
Combinatorics and Coffee Club, held at the Worldwide Center of Mathematics in
"... The event was organized by Jim Propp. Darij Grinberg scribed these notes. The descriptions of problems below are not direct quotations; they are meant to summarize each presentation (and are certainly subject to my misunderstanding and ignorance). My concentration during the actual meeting was at a ..."
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The event was organized by Jim Propp. Darij Grinberg scribed these notes. The descriptions of problems below are not direct quotations; they are meant to summarize each presentation (and are certainly subject to my misunderstanding and ignorance). My concentration during the actual meeting
a b c
"... Let a, b, c be positive real numbers. Prove that a 3 + abc (b + c) 2 + b3 + abc (c + a) 2 + c3 + abc 3 2 (a + b) 2 · a3 + b3 + c3 a2 + b2. + c2 Solution by Darij Grinberg. We will use the ∑ sign to denote cyclic summation (i. e., summation over the cyclic transpositions of our 3 variables a, b, c). ..."
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Let a, b, c be positive real numbers. Prove that a 3 + abc (b + c) 2 + b3 + abc (c + a) 2 + c3 + abc 3 2 (a + b) 2 · a3 + b3 + c3 a2 + b2. + c2 Solution by Darij Grinberg. We will use the ∑ sign to denote cyclic summation (i. e., summation over the cyclic transpositions of our 3 variables a, b, c
unknown title
"... Abstract. We give a short proof of Lemoine’s theorem that the Lemoine point of a triangle is the unique point which is the centroid of its own pedal triangle. Lemoine’s theorem states that the Lemoine (symmedian) point of a triangle is the unique point which is the centroid of its own pedal triangle ..."
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Yiu [8, §4.6.2]. Darij Grinberg [3] has given a synthetic proof. In this note we give a short proof by applying two elegant results on orthologic triangles. Lemma 1. If P is a point in plane of triangle ABC, with pedal triangle A′B′C ′, then the perpendiculars from A to B′C ′, from B to C ′A′, from C
Lifting The Exponent Lemma (LTE)
, 2011
"... Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It is pretty wellknown in the Olympiad folklore (see, e.g., [3]) though its origins are hard to trace. Mathematically, it is a close relative of the classical Hensel’s lemma (see [2]) in number theory (in ..."
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elementary mathematics. Understanding the theorem’s usage and its meaning is more important to you than remembering its detailed proof. I have to thank Fedja, darij grinberg(Darij Grinberg), makar and ZetaX(Daniel) for their notifications about the article. And I specially appreciate JBL(Joel) and Fedja
Recommended reading: [7] G. D. James, Representation theory of the symmetric groups, Springer Lecture Notes in Mathematics 692, Springer (1980).
"... Acknowledgement. I thank Darij Grinberg for corrections to numerous errors in the original version of these notes, and also for supplying many useful clarifications and suggestions for extra content. I thank Bakhshinder Samra for further corrections. Of course I take full responsibility for all th ..."
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Acknowledgement. I thank Darij Grinberg for corrections to numerous errors in the original version of these notes, and also for supplying many useful clarifications and suggestions for extra content. I thank Bakhshinder Samra for further corrections. Of course I take full responsibility for all
Piecewiselinear and birational toggling
"... Abstract. We define piecewiselinear and birational analogues of toggleinvolutions, rowmotion, and promotion on order ideals of a poset P as studied by Striker and Williams. Piecewiselinear rowmotion relates to Stanley’s transfer map for order polytopes; piecewiselinear promotion relates to Schü ..."
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Cited by 3 (0 self)
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to Schützenberger promotion for semistandard Young tableaux. When P = [a] × [b], a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewiselinear rowmotion) is of order a + b. We prove some homomesy results, showing that for certain functions
C'
"... Problem. Let G be the centroid of a triangle ABC; and let g be a line through the point G: The line g intersects the line BC at a point X: The parallel to the line BG through A intersects the line g at a point Xb: The parallel to the line CG through A intersects the line g at a point Xc: ..."
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Problem. Let G be the centroid of a triangle ABC; and let g be a line through the point G: The line g intersects the line BC at a point X: The parallel to the line BG through A intersects the line g at a point Xb: The parallel to the line CG through A intersects the line g at a point Xc:
unknown title
"... For any positive integer n, we define an integer n!! by n!! = k∈{1,2,...,n}; k≡n mod 2 ..."
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For any positive integer n, we define an integer n!! by n!! = k∈{1,2,...,n}; k≡n mod 2
EHRMANN’S THIRD LEMOINE CIRCLE
"... Abstract. The symmedian point of a triangle is known to give rise to two circles, obtained by drawing respectively parallels and antiparallels to the sides of the triangle through the symmedian point. In this note we will explore a third circle with a similar construction — discovered by JeanPierre ..."
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Abstract. The symmedian point of a triangle is known to give rise to two circles, obtained by drawing respectively parallels and antiparallels to the sides of the triangle through the symmedian point. In this note we will explore a third circle with a similar construction — discovered by JeanPierre Ehrmann [1]. It is obtained by drawing circles through the symmedian point and two vertices of the triangle, and intersecting these circles with the triangle’s sides. We prove the existence of this circle and identify its center and radius. 1. The first two Lemoine circles Let us remind the reader about some classical triangle geometry first. Let L be the symmedian point of a triangle ABC. Then, the following two results are wellknown ([3], Chapter 9):
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