Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On the Counting of Fully Packed Loop Configurations: Some new conjectures. | On the Counting of Fully Packed Loop Configurations Some new conjectures . Zuber Service de Physique Theorique de Saclay CEA DSM SPhT Unite de recherche associee au CNRS F-91191 Gif sur Yvette Cedex France zuber@ Submitted Nov 25 2003 Accepted Jan 27 2004 Published Feb 14 2004 MR Subject Classifications Primary 05A19 Secondary 52C20 82B20 Abstract New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain for periodic boundary conditions and so-called identified connectivities up to size 2n 22. 1. Introduction Fig. 1 The n X n grid here n 3 and n 4 with 2n external links occupied Consider a n X n square grid with its 4n external links see Figure 1. We are interested in Fully Packed Loops FPL in short . sets of disconnected paths which pass through each of the n2 vertices of the grid and exit through 2n of the external links every second of them being occupied see figure 2 for the case n 4 . There is a simple one-to-one correspondence between such FPL and alternating-sign matrices ASM obtained as follows divide the n2 vertices into odd and even as usual and associate 1 resp. 1 to each horizontal segment of the path passing through an even resp. odd vertex the opposite if the segment is vertical and 0 if the path has a corner at that vertex. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R13 1 A- -. 7 IT- TP- w T i r- n LPr- n 1 1 rL 1 h L Jn Ju c J J c r r I I I I I I I 9 Fig. 2 The 42 FPL configurations on a 4 X 4 grid. Configurations corresponding to distinct link patterns are separated by semi-colons. Fig. 3 FPL-ASM correspondence This prescription associates an n X n ASM matrix to the FPL configuration in a one-to-one way. Thanks to the celebrated result on ASM s 1 2 the total number of FPL is thus known to be An nf 11 For a review .
