A simple explicit bijection between ( n , 2 ) Gog and Magog - PowerPoint PPT Presentation

6 6 6 6 6 6 6 6 6 Introduction The bijection A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids Jérémie B ETTINELLI March 7, 2016 Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

5 6 6 6 6 6 6 6 6 Introduction The bijection What are Gog and Magog? Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

5 6 6 6 6 6 6 6 6 Introduction The bijection What are Gog and Magog? In the mathematical world, these are combinatorial objects known to be in bijection with other fundamental objects. Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 5 6 6 6 6 6 6 6 Introduction The bijection Alternating sign matrices Definition An alternating sign matrix of size n is an n × n matrix with entries in {− 1 , 0 , 1 } such that, on each fixed row or column, the nonzero entries start and end by 1 and alternate between 1 and -1.  0 1 0 0  1 − 1 1 0     0 1 0 0   0 0 0 1 Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 Alternating sign matrices Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 -1 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 -1 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 -1 1 0 0 1 0 0 1 -1 0 1 0 0 0 0 1 0 -1 1 0 0 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 6-vertex model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog loop model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog even coordinates odd coordinates loop model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog loop model Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 -1 1 0 0 0 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 0 1 0 0 1 0 0 1 0 0 0 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 0 1 0 0 1 -1 1 1 0 0 1 0 0 1 0 0 1 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 1 1 0 0 0 5 1 1 1 0 0 4 5 3 4 4 1 0 1 1 0 2 2 3 3 1 1 1 1 2 1 1 0 0 1 1 1 1 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 5 6 6 6 6 6 6 Introduction The bijection Gog 1 1 0 0 0 5 1 1 1 0 0 4 5 3 4 4 1 0 1 1 0 2 2 3 3 1 1 1 1 2 1 1 0 0 1 1 1 1 1 0 Gog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Totally symmetric self-complementary plane partitions Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Totally symmetric self-complementary plane partitions Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Totally symmetric self-complementary plane partitions Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Non intersecting lattice paths Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Non intersecting lattice paths Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Non intersecting lattice paths Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog Non intersecting lattice paths Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog + − − + − + + − − − + − − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog + − − + − + + − − − + − − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog + − − + − + + − − − + − − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog + − − + − + + − − − + − − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog 1 1 2 3 4 4 1 1 2 2 3 − 1 1 1 1 + − 1 1 1 + + − − − + − 1 1 1 − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016

6 6 6 5 6 6 6 6 6 Introduction The bijection Magog 1 2 3 4 5 6 1 1 2 3 4 4 1 1 2 2 3 − 1 1 1 1 + − 1 1 1 + + − − − + − 1 1 1 − + − − − Magog Jérémie B ETTINELLI A simple explicit bijection between ( n , 2 ) Gog and Magog trapezoids March 7, 2016