Dan Li - cp's OEIS frontend

User: Dan Li

Dan Li has authored 11 sequences. Here are the ten most recent ones:

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

Dan Li, Apr 27 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

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.

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407, A383408.

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)}).

Original entry on oeis.org

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

Dan Li, Apr 27 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

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.

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407, A383408, A383433.

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.

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)}).

Original entry on oeis.org

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

Dan Li, Apr 26 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

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.

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406.

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)}).

Original entry on oeis.org

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

Dan Li, Apr 26 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

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.

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407.

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.

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)}).

Original entry on oeis.org

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

Dan Li, Apr 25 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.4 at page 17.

Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312.

G.f.: (1 + t)^2 *A(t)/((1 + t)^2 + t^2*(A(t) - 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.

A383107 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).

Original entry on oeis.org

1, 1, 0, 0, 2, 14, 88, 636, 5174, 47122, 475128, 5257976, 63381078, 826817350, 11607019144, 174484968604, 2796703640190, 47613279070594, 858080079253440, 16320196781972904, 326687694661023774, 6865553778933359142, 151139392725808178080, 3478151644016630307452, 83516524547918673461238

Offset: 0

Dan Li, Apr 22 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.2 at page 15.

Cf. A002464, A382644, A382645, A382651, A383040.

G.f.: (1/(1 + t) + t*(1 + t)/(1 + t + t*A(t)))*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.

A383040 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(2,0)}).

Original entry on oeis.org

1, 1, 0, 0, 2, 12, 78, 568, 4674, 42944, 436314, 4860020, 58914870, 772330276, 10888803374, 164310553184, 2642525580218, 45124440536632, 815438318526482, 15547317496485932, 311912067538692126, 6568399090178800988, 144867849880285518694, 3339550150164041194232, 80315480372245746015970

Offset: 0

Dan Li, Apr 22 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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: 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.1 at page 13.

Cf. A002464, A382644, A382645, A382651.

G.f.: (1 + t)^2*A(t)/(1 + t + t*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.

A383312 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,1),(2,0),(2,2)}).

Original entry on oeis.org

1, 1, 0, 0, 2, 14, 86, 624, 5096, 46554, 470446, 5214936, 62943852, 821949042, 11548027442, 173711893048, 2785807179384, 47448884653218, 855436571437710, 16275060021803232, 325872090863707740, 6850004083354211050, 150827444158572339810, 3471582648001267649808, 83371646323922972242776

Offset: 0

Dan Li, Apr 22 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.3 at page 16.

Cf. A002464, A382644, A382645, A382651, A383040, A383107.

G.f.: (t + 1/(1 + t) - t^2*A(t)^2/((1 + t)*(1 + t + t*A(t))))*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.

A382645 Number of king permutations on n elements not beginning with the smallest element and not ending with the largest element.

Original entry on oeis.org

1, 0, 0, 0, 2, 10, 68, 500, 4174, 38774, 397584, 4462848, 54455754, 717909202, 10171232060, 154142811052, 2488421201446, 42636471916622, 772807552752712, 14774586965277816, 297138592463202402, 6271277634164008170, 138596853553771517492, 3200958202120445923684, 77114612783976599209598

Offset: 0

Dan Li, Apr 01 2025

nonn

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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, and 53142.

Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See formula (3) at page 5.

Cf. A002464, A382644.

nmax = 20; CoefficientList[Series[x/(1+x) + Sum[k!*x^k*(1-x)^k/(1+x)^(k+2), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 04 2025 *)

my(N=30, t='t+O('t^N)); Vec(t/(1+t)+sum(n=0,N,n!*t^n*(1-t)^n/(1+t)^(n+2))) \\ Joerg Arndt, Apr 03 2025

G.f.: t/(1+t) + Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^(n+2).

a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Apr 04 2025

A382644 Number of king permutations on n elements not beginning with the smallest element.

Original entry on oeis.org

1, 0, 0, 0, 2, 12, 78, 568, 4674, 42948, 436358, 4860432, 58918602, 772364956, 10889141262, 164314043112, 2642564012498, 45124893118068, 815444024669334, 15547394518030528, 311913179428480218, 6568416226627210572, 144868131187935525662, 3339555055674217441176, 80315570986097045133282

Offset: 0

Dan Li, Apr 01 2025

A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0

			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: 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, and 53142.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Dan Li and Philip B. Zhang, Distributions of mesh patterns of short lengths on king permutations, arXiv:2411.18131 [math.CO], 2024. See formula (2) at page 5.

Cf. A001266, A002464, A382645.

a:= proc(n) option remember; `if`(n<4, [1,0$3][n+1],
      (n+1)*a(n-1)-(n-3)*(a(n-2)+a(n-3))+(n-2)*a(n-4))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 04 2025

nmax = 20; CoefficientList[Series[Sum[k!*x^k*(1-x)^k/(1+x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 04 2025 *)

my(N=30, t='t+O('t^N)); Vec(sum(n=0,N,n!*t^n*(1-t)^n/(1+t)^(n+1))) \\ Joerg Arndt, Apr 03 2025

G.f.: Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^(n+1).
a(n) = (n+1)*a(n-1) - (n-3)*(a(n-2)+a(n-3)) + (n-2)*a(n-4) for n>=4. - Alois P. Heinz, Apr 04 2025
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Apr 04 2025

cp's OEIS Frontend

User: Dan Li

A383433 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,2),(2,1)}).

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A383434 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,2),(1,0),(1,1),(2,0),(2,1)}).

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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)}).

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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)}).

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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)}).

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A383107 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)}).

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A383040 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(2,0)}).

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A383312 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,1),(0,2),(1,0),(1,1),(2,0),(2,2)}).

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A382645 Number of king permutations on n elements not beginning with the smallest element and not ending with the largest element.

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