V. Reiner "Signed Posets" R. Stanley "A Symmetric - PDF document

Resourses V. Reiner "Signed Posets" R. Stanley "A Symmetric Generalization of the Chromatic Polynomial of a Graph" T . Zaslavsky "Signed Graph Coloring" R. Adin et al. "Character Formulas and Descents for the Hyperoctahedrial Group" Air Hannah Johnson Eric Fawcett Mikey Reilly Kat Husar An example to give intuition about Stanley 3 I Proof 3.3 of posets Root systems signed posets descent sets and linear extensions

What done so far have we Stanley's theorem for a graph 6 with n verticies 11h X of acyclic orientations f 1 of G for generalization generalization chromatic for sighed symetric graphs Generalization polynomial with signed graph Bf BARF and B symetric chromatic Zaslavski's Theorem Theorem Z Stanley's polynomial for a signed graph 2 with n verticies Xg c e n XE f 1 f 1 of acyclic XG symetric chromatic orientations of G elementary symetric function e E aj G Cy Xid la generalization for signed graphs generalization for B symetric chromatic polynomial ppg Whk 7700 trying to find what we are G of acyclic aj orientations with j sinks l 7 of sets created by partition

Preview ApApApApApAp d ol it i I n r innumerate ETE Quasi Symmetric function Xnah FCK Xia Xie nah FCx x x where i Liza din jzL j Ljn F x f x xz xzXy X xzxztxfxzxytxixzxytxix.ly Quasi basis Symmetric Qs d x _Qs X d 14 11,2 d H E Xi Xie where S is a subset of lid i s Eid ijcijtlifj.ES let 5 12,33 Ex d 13 11,2 33 let D 4 is yiiiHi3Xiy E zEjcYjzXis'iy xfXzXzXytX xzxsx.oo Qs y iz c iz iz Liu

Bo The convention for the rest of my presentation a b b a Poset a partial order linear extension a totalorder that preserves F x the orderfrom the c 3rd s 3rd c E s 3rd F poset f s j E f S linear extension as c E a permutation E I sac E f 43yd C S D 3 Y 5 Z f linear extension as represent this poset as a picture a picture o F f c tog f g 3rd 3rd 3 4 l S Z for 1 graph there is only 1 poset ftp.x There can be 1 to many linear extensio Order reversing linear extension Descent set Stanley D D Ej taj f s 3rd E c a t wca uns father w fI YE's twin t lad 3,5 4 2,1 F la az az ay as 7 c f j 3,5 4,2 l s 3h 2 3 4 55 j 1 4 1 4233 2 3 2 3,43 1 d You choose 1 two linear extensions can have the same descent w think of descent sets as contradictions between a few You can 1 5 3 Z y s

823 LCP the set of all linear extensions w defines the following Stanley xp th a Ep d with the orientation 1 Theorem 3 I Q Dia E Xp XEL P w 😖 y E Xp Xun x XKcvd compatible with the orientation look familiar E Xi X zoo lid Qs Ct Xp E Xkcy Xp x cry i E Eid Ktv EklVz ijcijtlifj.ES EklVd if vz S AV then Valued Kcr F F x y c f j w a e o d S 3 3rd 7 5 4 3 2 J a b s o k be C y compatible coloring f y Fog K 7 POSET f c 3rd i'm goin to color this b graph c 9 colors a with Combine K to create and w 2 that are a unique compatible c f have three colors a b c we of DCL 11,31 b c a y 3 4 I 21 5 a

Theorem 3.3 symetric chromatic lg fyCxe7 of acyclic with j aj G let orientations sinus E G c g Itd lCN j has 6 acyclic F x triangle with 1 orientations sink 3 ez Xo Proof an acyclic orientation of G 0 V L is K is a proper coloring G compatible 8 compatible if K is O o b 3 a V Ca Kcb 3 N L Every proper coloring is compatible with acyclic orientation z one L Z a L ko o Ocompatible proper colorings k o Ng E all colorings proper Ko Voko In o Eo o Eo Xo Xo I X af a E closure of 0 is the transitive O o

0 notice that a poset 0 is since is acyclic o X g Xp KO KO since Tx a y off Eg Xf Thus XG f CX tfCy flex cfc linear transformation f City 0 t t Magic try Qsy if 5 Eiti ttt 11 d il it f O otherwise P y tm for any d element 4 Xp claim poset of minimal elements number m e Proof of Claim r 9 h order reversing bijection is w f steps to get linear extension d b d Ca adl with descent set iti il z d 13 V is the minimal element of P 1 with largest wcv value R 2 anyi minimal elements N choose of P that are not v g list in increasing order of tables m 4 3 list C v remaining elements of P list 4 a order of tables in poset P decreasing 6 e i possibleelineffsions descent sets 4 31 1 5678432 I i 3 4,5 6,73 h w i _z 6785432 I 3,4 5 6,73 Y 1 489 3 57864321 3,4 s 6,73 i 2 5 d z 56874321 E3 4,5 6,73 i Z 5876432 I 2,3 4 5,673 V i _I 41 1 3 68754321 2,3 y 5,673 l i A 78654321 i i 2,3 415,673 4 1 87654321 9 1 o 91,2 3,4 5,673 i O 7 yo z A 12345678

7 for There choices u are u Thus ycxpi.FI 7 It t Ili E Q Dix since Xp by 3.1 LLP w hat follow unique descent sets 5 Eiti ite and there are in d is linear extension mf and corresponding i o t t l if s Eiti d if it y g 0 otherwise we made it YtCt lli tm D8 QCxpi E.FI Lets finish up 3.3 3 Eg Xf XG the number of 0 is acyclic then if minimal elements the number of sinks is 61 418 Colt 01 41 a Eff Caen the poset which let is disjoint Py be of chains union X of cardinalities Xz Tj fl thus 46,1 Thus eyez _ex Xp 1 3 I 12314567819103 X JI 72 5 73 2 3 X s

a extent d 4 fat cat a g CG fed Cx lCH j