A383406 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)}).
1, 1, 0, 0, 2, 14, 88, 632, 5152, 46976, 474056, 5249064, 63298724, 825977620, 11597642568, 174371083288, 2795208188972, 47592162832412, 857760977798888, 16315057829100968, 326599827759568812, 6863964030561807340, 151109048051281532488, 3477542225297684400056, 83503678542689445133052
Offset: 0
A383407 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).
1, 1, 0, 0, 2, 14, 88, 636, 5174, 47122, 475124, 5257936, 63380706, 826813990, 11606987816, 174484661916, 2796700455414, 47613243806514, 858079661762692, 16320191491499712, 326687622910353650, 6865552738575268502, 151139376627154723752, 3478151378775992816412, 83516519907235226131286
Offset: 0
Comments
A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0
Examples
For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142. For n = 5 the a(5) = 14 solutions are these 14 permutations: 13524, 14253, 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
Links
- Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.5 at page 18.
Formula
G.f.: (1 + t)*(1 + t + t*(2 + t)*A(t))*A(t)/(1 + t + t*A(t))^2 where A(t)=Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
A383408 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,1),(1,2),(2,1)}).
1, 1, 0, 0, 2, 14, 88, 632, 5152, 46972, 474008, 5248616, 63294680, 825940168, 11597278752, 174367336624, 2795167052832, 47591679875632, 857754907053056, 16314976128578752, 326598651690933216, 6863945954213702816, 151108752072042907968, 3477537076217415673344, 83503583639127861347392
Offset: 0
Comments
A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0
Examples
For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142. For n = 5 the a(5) = 14 solutions are these 14 permutations: 13524, 14253, 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
Links
- Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.6 at page 19.
Formula
G.f.: (1 + t)*(A(t) - t)/(1 + t*(A(t) - t - 1)) where A(t)=Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
A383433 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,2),(2,1)}).
1, 1, 0, 0, 2, 12, 76, 556, 4592, 42328, 431184, 4812936, 58440200, 767098296, 10826066776, 163496360680, 2631146363384, 44953977477160, 812713132751832, 15501004918724712, 311078390317974872, 6552553451281418472, 144550752700158416920, 3332886257051337065128, 80168754370190239256408
Offset: 0
Comments
A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0
Examples
For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142. For n = 5 the a(5) = 12 solutions are these 12 permutations: 13524, 14253, 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
Links
- Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.7 at page 21.
Crossrefs
Formula
G.f.: (2*A(t) - t - 1)/(A(t) - t) where A(t)=Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
A383434 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,2),(1,0),(1,1),(2,0),(2,1)}).
1, 1, 0, 0, 2, 10, 68, 500, 4170, 38730, 397172, 4459116, 54421082, 717571442, 10167743668, 154104395348, 2487968793386, 42630767594522, 772730550801940, 14773475294401180, 297121458577213850, 6270996358146824738, 138591948457411817684, 3200867594024256790020, 77112844928711640695594
Offset: 0
Comments
A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0
Examples
For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142. For n = 5 the a(5) = 10 solutions are these 10 permutations: 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
Links
- Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.8 at page 23.
Crossrefs
Formula
G.f.: 1 + t + 1/(1 + t) - 1/A(t) where A(t)=Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
Comments
Examples
Links
Crossrefs
Formula