# Classes of S6 according to pattern τ avoiding involutions and the numbers #I11(τ) and #I12(τ).

Let Sn be the set of all permutations of 1,2,...,n (permutations of length n). An involution a1a2...an is a permutation of length n such that aai=i for all i=1,2,...,n. A pattern is a permutation. Two patterns τ and υ are belong to the same I-Wilf class if and only if the number of involutions in Sn that avoid τ is the same as the number of involutions in Sn that avoid υ, for all n≥0, that is,

### #In(τ)=#In(υ).

In S6, the set of permutations of length 6, there are 6!=720 permutations. In the table below presented the all the I-Wilf classes of patterns in S6, 203 classes.

Pattern τ #I11(τ) #I12(τ) Pattern τ #I11(τ) #I12(τ) Pattern τ #I11(τ) #I12(τ) Pattern τ #I11(τ) #I12(τ)
3615422746797405 4651322747997511 3614522773498805 3516242779199133
4261532780999287 1462532778799321 1325462780199432 1254362783399521
1543262783899585 1536242784899650 1243562784999653 1235462785699729
6243512787499857 6254312789899885 1234562790899991 6235412791310002
6452312791910008 6325412792810015 5634122796310029 6234512800010061
1635422807510087 4631522810810099 1643522813010119 1256342817610140
1564232818210145 1452362822810166 1264532823610175 1634522825510191
1534262829810210 1354262928410423 1365422951310531 1246532959410597
1245362974510678 1543622979610685 1563422983210718 1254632990310757
3261542994210777 1345263019910808 1362543002810833 2654313019010896
1436253014110896 1453263042410929 2615433023910940 1436523048310944
4625133025210951 1325643026810967 1352463033410994 1364523060411013
1235643037611026 1346523077411070 1245633071211087 1354623082511096
1463523068611102 1435623087511122 6354213057411159 2643513090811164
1356243084311164 2635413092011173 1534623090711183 1246353066511187
3625413090411196 1256433091111205 6245313068111218 4625313075711223
1564323102511249 2614533101311259 1536423101011273 2536143108411280
1452633104211283 2461533110011296 1346253108411303 3265413120111310
1345623122511312 4632513114411315 2361543114311316 2634513122611333
3624513119311342 1645323092811343 1546233120411369 1365243124211383
4265133125111390 1362453128911404 3516423130911406 2365413138511407
2543613138911412 4623513135311424 1463253135811447 2563413146111459
3265143143311473 1465233142011483 1465323146211505 3641523146411505
5624313149911513 2516343147311516 4635123149311528 5643213153711529
2613543156211530 2436153158811535 2645133158311550 3651423149811553
3246513167111560 6352413128011560 2564133162911571 2436513170411574
2641533156711576 6345213155311601 5642313158311608 1546323165011609
2645313168111620 3654213168011621 2654133177011654 2416533176411658
2346513183111660 1356423202311665 1453623202411666 5623413173611667
2365143182811668 2354613186611674 2513643177511700 6453213174711719
4653123184911734 2346153198711753 1352643193011764 2345613199411766
3256143221511779 2563143214111836 2651433201511837 2315643226611845
2316453227511851 3461523207411853 5634213229011864 3264513245211872
1456233211111888 4653213217211904 2643153251511908 2465133250611920
1364253246011926 2516433224611928 2361453247111930 2615343248311941
2564313256311948 4265313248611959 2561343225211974 2364513265211986
4563123232112002 3564123235012004 3561423236612019 3642513244312026
2356143269212027 2546133270712043 2653413268912045 3625143269412065
2534613282512079 2463513279012092 2546313280212102 3654123277812107
2463153278912112 4652313275812128 2631543289712134 1456323290512139
2635143290512157 2514633286912169 2541633289612169 2351643289612171
2536413295812178 2634153304912189 3256413301512193 2461353292212195
2465313301412212 3562413302612242 2451633308412242 4263513306212245
2561433302712248 4365123310012260 2416353300212266 3645213305212272
3526413311312284 2356413317112289 2456133314512295 2453613322912319
3462513322512325 4635213315012337 4652133312312341 4561323315012347
3645123322412351 4562313318612375 2364153335912383 3562143328512383
3546213332312393 3652413328112419 3465123337412440 3561243339812493
2651343351312505 2653143359412554 2456313369312566 3652143354512573
3564213376012625 3456123365012626 4365213373812655 3465213381512674
3546123381712701 4563213398812759 3456213421912880