# Classes of words τ of size 4,5 according to pattern τ avoiding compositions with parts in N={1,2,3,...}.

Let [k]m be the set of all words π=π1π2…πm on the alphabet {1,2,...,k} of length m. A composition of n with parts in N={1,2,...} is a word π=π1π2…πm of length m such that π12+…+πm=n. We say k-ary word π=π1π2…πnof length n contains a k'-ary word τ of length n' if there exists a subsequence 1≤i1<…<in'≤n such that πi1πi2…πin' is order-isomorphic to τ. Otherwise, we say π avoids τ. We denote the set of all compositions of n with parts in N that avoid τ by Cn(τ). Two words τ and τ' are equivalent if there is a size-preserving bijection between the sets Cn(τ) and Cn(τ').

Patterns of size 4:
• |Cn(1112)|n=0...24=|Cn(1121)|n=0...24
=1,1,2,4,8,15,29,54,101,185,335,597 ,1067,1872,3257,5626 ,9655 ,16374,27655 ,46240 ,77075 ,127434,209453 ,341764 ,555698.

• |Cn(1111)|n=0...24
=1,1,2,4,7,15,26,52, 93,173,310,556 ,1041,1789,3098,5620 ,9725 ,16377,28764 ,48518 ,82889 ,137161,237502 ,390084 ,646347.

• |Cn(2112)|n=0...24
=1,1,2,4,8,16,31,61,118,228,433,819 ,1536,2858,5283,9679 ,17621,31811,57081 ,101691,180111,316951,554561 ,964485 ,1668026.

• |Cn(1212)|n=0...24
=1,1,2,4,8,16,31,61,118,228,433,819 ,1535,2857,5280,9675 ,17609,31791,57043 ,101638,180057,316955,554777 ,965283 ,1670341.

• |Cn(1221)|n=0...24
=1,1,2,4,8,16,31,61,118,228,433,819 ,1536,2860,5289,9695 ,17659,31909,57319 ,102253,181381,319715,560407 ,976597 ,1692732.

• |Cn(1122)|n=0...24
=1,1,2,4,8,16,31,61,118,228,433,819 ,1538,2866,5303,9729 ,17741,32105,57767 ,103235,183465,324009,569113 ,993995 ,1726932.

• |Cn(1312)|n=0...24
=1,1,2,4,8,16,32,63,124,241,466,892 ,1694,3191,5962,11054,20340,37161,67418 ,121524,217661,387562,686136 ,1208267,2116810.

• |Cn(1231)|n=0...24
=1,1,2,4,8,16,32,63,124,241,466,892 ,1694,3192,5967,11072,20395,37315,67813 ,122485,219897,392573,697020 ,1231297,2164445.

• |Cn(2113)|n=0...24
=1,1,2,4,8,16,32,63,124,241,466,892 ,1694,3193,5970,11081,20420,37378,67960 ,122823,220639,394149,700295 ,1237957,2177740.

• |Cn(1213)|n=0...24
=1,1,2,4,8,16,32,63,124,241,466,892 ,1694,3193,5970,11082,20424,37392,68004 ,122946,220961,394948,702193 ,1242302,2187393.

• |Cn(1123)|n=0...24=|Cn(1132)|n=0...24
=1,1,2,4,8,16,32,63,124,241,466,892 ,1694,3193,5971,11086,20436,37427,68096 ,123170,221485,396128,704754 ,1247712,2198554.

• |Cn(1222)|n=0...24=|Cn(2122)|n=0...24
=1,1,2,4,8,16,32,63,124,244,477,927 ,1793,3451,6607,12578,23824,44909,84261 ,157331,292357,540667,995338 ,1824181,3328584.

• |Cn(1223)|n=0...24=|Cn(1232)|n=0...24=|Cn(1322)|n=0...24=|Cn(2123)|n=0...24=|Cn(2132)|n=0...24=|Cn(2213)|n=0...24
=1,1,2,4,8,16,32,64,127,251,494,967 ,1882,3642,7006,13401,25490,48219,90722 ,169789,316127,585660,1079755,1981407,3619549.

• |Cn(2313)|n=0...24
=1,1,2,4,8,16,32,64,128,255,508,1009,2001,3957,7808,15364,30154,59019,115202,224253,435348,842864,1627482,3134190,6020078.

• |Cn(1332)|n=0...24
=1,1,2,4,8,16,32,64,128,255,508,1009,2001,3957,7808,15364,30154,59021,115210,224285,435447,843157,1628279,3136275,6025299.

• |Cn(1323)|n=0...24
=1,1,2,4,8,16,32,64,128,255,508,1009,2001,3957,7808,15364,30154,59021,115210,224285,435449,843165,1628312,3136381,6025619.

• |Cn(3123)|n=0...24
=1,1,2,4,8,16,32,64,128,255,508,1009,2001,3957,7808,15364,30155,59024,115222,224319,435546,843417,1628950,3137925,6029274.

• |Cn(1233)|n=0...24=|Cn(2133)|n=0...24
=1,1,2,4,8,16,32,64,128,255,508,1009,2001,3957,7808,15364,30155,59025,115226,224334,435591,843544,1629284,3138770,6031329.

• |Cn(2413)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61181,120014,234818,458273,892134,1732500,3356468,6487681.

• |Cn(1423)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61181,120014,234819,458278,892154,1732568,3356678,6488279.

• |Cn(1342)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61181,120014,234819,458278,892156,1732578,3356721,6488429.

• |Cn(2314)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61182,120019,234839,458346,892365,1733176,3358335,6492597.

• |Cn(3124)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61182,120019,234839,458346,892366,1733182,3358362,6492696.

• |Cn(1234)|n=0...24=|Cn(1243)|n=0...24=|Cn(1432)|n=0...24=|Cn(2134)|n=0...24=|Cn(2143)|n=0...24=|Cn(3214)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61182,120019,234840,458351,892386,1733249,3358567,6493277.

• |Cn(1324)|n=0...24
=1,1,2,4,8,16,32,64,128,256,511,1019,2028,4028,7981,15777,31108,61182,120019,234840,458351,892387,1733255,3358594,6493377.

Patterns of size 5:
Pattern τ |C20(τ)| |C21(τ)| |C22(τ)| |C23(τ)| |C24(τ)| |C25(τ)| |C26(τ)| |C27(τ)| |C28(τ)| |C29(τ)| |C30(τ)| |C31(τ)| |C32(τ)|
11112,11121,11211170944299744 522829 905728 1561794
11111170015303356 536877 932679 1637383
12112277374506651 920031 16615322984319
21112277480506816 920330 16619162984760
11212278222508621 924526 16714573005810
12121278276508768 924892 16723463007872
11221280120512922 933962 16917503048552
11122280794514454 937414 16992503064692
13112317119585160 107358719587833554467
11312317346585816 107537219633463565653
12131317628586559 107722619677863575872
11231318174587969 108069419759873594640
21113318202588047 108091819765503595962
11321318264588201 108127519773783597877
12113318265588243 108147919780553599799
11213318412588642 108249919805533605700
11123,11132318804589632 108487219860543618082
21212369350698204 131404724624404594680
12122369389698312 131433824631814596534
12212369381698303 131432724631964596646
12221369544698720 131535924656904602480
21122369782699234 131642124677844606410
11222370136700162 131878124735904620294
21312392795744739 140541026399214936338
13212392822744825 140565826405894938041
13122393012745249 140655526424014941526
21132392964745170 140646826424204942050
12312392999745302 140688026435934945188
21213393008745334 140697726438614945872
12132393010745344 1407015264398749462519212699
12123393010745344 1407015264398749462519212703
21123393055745435 140718526442564946564
12231393092745510 140732426445004946943
12213393154745715 140793626461954951406
12321393159745731 140798026463074951672
22113393413746377 140955626500214960144
11223,11232,11322393429746443 140977826506994962054
12222,21222,22122428952827634 159163230509655829342
12223,12232,12322
13222,21223,21232445647861514 165955131854666092743
21322,22123,22132,22213
13132472809924977 180512235140116823654
21313472818925013 180523135143376824536
12313472823925037 180533135146716825565
13123472823925038 180534135147136825727
13213472825925047 180537635148296826077
13231472847925107 180553935152356827061
13312472854925129 180559635153736827360
12331472857925146 180566835156226828139
23113472903925247 180587535159936828696
31123472918925309 180608635166516830599
31213472929925355 180622935170856831797
11332472947925405 180638335175016832927
11323472947925405 180638235175066832945
21133472963925467 180656935180476834381
12133472963925467 180657435180696834471
11233472963925471 180659335181396834703
13142488636960244 188277036832167189065
14213488638960253 188280636833397189448
21413488638960254 188281036833547189492
12413488638960255 188281636833837189608
13412488639960260 188283936834617189854
14123488643960272 188287236835437190038
14312488640960265 188286136835427190126
21314488640960267 188287436836027190350
12431488644960278 188289836836407190366
13124488640960267 188287436836047190369
13241488644960279 188290236836537190403
14132488645960283 188291636836987190536
24113488674960372 188316036843217192033
12314488665960355 188315236844147192588
13214488665960355 188315436844257192639
12341488666960358 188316136844377192641
31124488679960398 188327136847197193333
11423488682960408 188329736847857193473
11342488682960408 188329736847847193477
31214488680960405 188330236848377193723
21134,21143488680960405 188330436848477193765
12134,12143488681960410 188332436849177193987
23114488685960422 188335536849847194103
32114488685960422 188335836850007194171
11234,11243,11432488686960428 188338336850947194484
11324488686960428 188338336850947194488
23213494582974957 191810037659187378434
23132494584974968 191814737660827378953
23123494587974977 191816637661187378999
12323494587974979 191817737661667379166144277272814880754800750106456739206356820
21323494587974979 191817737661667379166144277272814880754800750106456739206356816
13232494587974979 191817737661667379168
31223494594975010 191828237664747379986
13223494594975009 191828037664747380013
32123494594975010 191828937665187380183
21332494594975012 191830037665677380361
12332494594975012 191830237665817380425
22313494598975033 191838837668627381265
13322494608975078 191854737673807382805
12233,21233,22133494608975080 191855937674347382995
24213504899998671 197181338860397644090
24123504899998671 197181338860397644095
13242504899998671 197181338860427644116150075642940672657508444
12423504899998671 197181338860427644116150075662940673657508486112244393218649086425095725824879229
21423504899998671 197181338860427644116150075662940673657508486112244393218649086425095725824879228
21342504899998673 197183038861327644491
23142504900998679 197185638862227644769
12342504900998680 197186238862507644877
21432504900998680 197186438862637644938
12432504900998680 197186438862637644939
14223504901998687 197189338863617645226
22413504901998687 197189538863757645294
14232504901998687 197189538863757645298
31224504901998687 197189738863877645347
32214504901998687 197189738863877645350
32124504901998687 197189738863877645352
23124504901998687 19718973886387764535315011590
23214504901998687 19718973886387764535315011589
13224504901998687 197189738863887645357
12324504901998687 197189738863897645366150116512941918057544352112343292218911964425773182826579262
21324504901998687 197189738863897645366150116512941918057544352112343292218911964425773182826579261
13422504901998688 197190138864037645401
22314504901998687 197189938864027645425
14322504901998688 197190438864247645504
12234,12243,21234
21243,22134,22143504901998688 197190538864307645532
233135119531017130201867940020397924995
231335119531017130201867940020397924996
313235119531017130201867940020397924998
1332351195310171302018679400203979250031567489230965800
1333251195310171302018679400203979250031567489230965797
132335119531017130201867940020397925004
312335119541017135201870040021147925241
321335119541017135201870040021147925242
12333,213335119541017135201870040021147925246
243135165891027944204343040577728048745
23413516589102794420434304057772804875015946331
3132451658910279442043430405777280487501594633431554879
2313451658910279442043430405777280487501594633431554899623636161230954952426553784777134419392254811844141265
2314351658910279442043430405777280487501594633431554899623636161230954952426553784777134419392254811844141264
314235165891027944204343140577778048767
2413351658910279442043432405778480488011594651431555463
1334251658910279442043432405778480488011594651431555465
1423351658910279442043432405778480488011594651431555467
134235165891027944204343240577848048804159465343155556362365649123101367242671568
134325165891027944204343240577848048804159465343155556362365649123101367242671567477720039
143325165891027944204343240577848048804159465343155556362365649123101367242671567477756437
2331451658910279442043432405778480488041594653531555566
1432351658910279442043432405778480488041594653531555569
1332451658910279442043432405778480488041594653531555570
3231451658910279442043432405778480488041594653531555580
1323451658910279442043432405778480488041594653531555582623657401231017232426728114777604359393482621844453674
1324351658910279442043432405778480488041594653531555582623657401231017232426728114777604359393482621844453673
312435165891027945204343640577988048843
321435165891027945204343740578048048869
3123451658910279452043437405780580488771594676731556254
3213451658910279452043437405780580488771594676731556256
3312451658910279452043438405781180489031594686131556255
3321451658910279452043438405781180489031594686131556258
12334,12343,12433
21334,21343,2143351658910279452043438405781180489031594686131556259
1424352214510426322081196415262282821981651086132899031
134245221451042632208119641526228282198165108633289904365520227
314245221451042632208119641526228282198165108633289904365520226
2414352214510426322081196415262282821981651086532899053
2413452214510426322081196415262282821981651086732899066
2431452214510426322081196415262282821981651086732899067
1434252214510426322081196415262282822001651087532899100
134425221451042632208119641526228282200165108773289911165520497
244135221451042632208119641526228282200165108773289911165520496
3241452214510426322081196415262382822051651089932899190
2341452214510426322081196415262382822051651089932899195
1423452214510426322081196415262382822061651090232899203
1432452214510426322081196415262382822061651090232899205
1442352214510426322081196415262382822071651090932899236
1443252214510426322081196415262382822071651090932899239
341245221451042632208119741526278282220165109413289929565520977
342145221451042632208119741526278282220165109413289929565520976
12443,214435221451042632208119741526278282222165109553289936765521272130420893
12434,214345221451042632208119741526278282222165109553289936765521272130420894
41324522145104263220811974152628828222716510978
42134522145104263220811974152628828222716510979
412345221451042632208119741526288282228
231445221451042632208119741526288282229165109913289951565521811130422680259467469515903007
312445221451042632208119741526288282229165109913289951565521811130422680259467469515903012
12344,21344,321445221451042632208119741526288282229165109913289951565521811130422684259467492515903117
132445221451042632208119741526288282229165109913289951565521811130422684259467492515903121
2415352319410453212087895416892483211001660217933110392
3152452319410453212087895416892483211001660217933110393
2531452319410453212087895416892483211001660217933110394
2451352319410453212087895416892483211001660217933110395
1425352319410453212087895416892483211001660218033110398
135245231941045321208789541689248321100166021803311040066003540131511362
251345231941045321208789541689248321100166021803311040066003540131511361
135425231941045321208789541689248321100166021803311040166003546131511390
143525231941045321208789541689248321100166021803311040166003546131511391
2541352319410453212087895416892483211001660218033110404
2351452319410453212087895416892483211011660218733110434
314255231941045321208789541689248321101166021873311043666003689
241355231941045321208789541689248321101166021873311043666003691
153245231941045321208789541689248321101166021883311044066003703
325145231941045321208789541689248321101166021883311044066003705
152345231941045321208789541689248321101166021883311044166003711
145235231941045321208789541689248321101166021883311044166003712131511996
145325231941045321208789541689248321101166021883311044166003712131511998
153425231941045321208789541689248321101166021883311044166003713
152435231941045321208789541689248321101166021883311044266003716
142355231941045321208789541689248321101166021883311044266003717
251435231941045321208789541689248321101166021883311044266003719131512031
154235231941045321208789541689248321101166021883311044266003719131512032
134255231941045321208789541689248321101166021883311044266003719131512033
324155231941045321208789541689248321101166021883311044266003719131512034
243155231941045321208789541689248321101166021883311044266003721
1345252319410453212087895416892483211011660218833110443
12534,215345231941045321208789541689258321106166022083311050766003908131512528261908366
12453,214535231941045321208789541689258321106166022083311050766003908131512528261908367
351245231941045321208789541689258321106166022073311049966003865131512342
352145231941045321208789541689258321106166022073311049966003865131512341
421355231941045321208789541689258321107166022153311054466004065131513117
413255231941045321208789541689258321107166022153311054466004065131513119
23145,231545231941045321208789541689258321107166022153311054566004072
31245,312545231941045321208789541689258321107166022153311054566004073
341255231941045321208789541689258321107166022163311055066004092131513218261910724521332661
342155231941045321208789541689258321107166022163311055066004092131513218261910724521332662
412355231941045321208789541689258321107166022163311055066004092131513218261910721
423155231941045321208789541689258321107166022163311055066004093
234155231941045321208789541689258321107166022163311055066004094131513234
431255231941045321208789541689258321107166022163311055066004094131513235
12345,12354,12543
15432,21345,213545231941045321208789541689258321107166022163311055166004100131513262261910916
21543,32145,32154,43215
12435,214355231941045321208789541689258321107166022163311055166004100131513262261910917
143255231941045321208789541689258321107166022163311055166004100131513263
13245,132545231941045321208789541689258321107166022163311055166004101