Jonathan Burns - cp's OEIS frontend

A271218 Number of symmetric linked diagrams with n links and no simple link.

1, 0, 1, 3, 12, 39, 167, 660, 3083, 13961, 70728, 355457, 1936449, 10587960, 61539129, 361182139, 2224641540, 13880534119, 90090083047, 593246514588, 4038095508691, 27905008440273, 198401618299920, 1432253086621377, 10600146578310209, 79639887325700592, 611739960145556273

Offset: 0

Jonathan Burns, Apr 13 2016

Number of symmetric chord diagrams (where reflection is equivalent) with n chords and no simple chords.

Number of symmetric assembly words that do not contain the subword aa.

			For n=0 the a(0)=1 solution is { ∅ }.
For n=1 there are no solutions since the link in a diagram with one link, 11, is simple.
For n=2 the a(2)=1 solution is { 1212 }.
For n=3 the a(3)=3 solutions are { 123123, 121323, 123231 }.
For n=4 the a(4)=12 solutions are { 12123434, 12132434, 12324341, 12314234, 12341234, 12342341, 12314324, 12341324, 12343412, 12343421, 12324143, 12342143 }.

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

Cf. A047974, A271215.

RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,20}]

lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); va;} \\ Michel Marcus, Jul 28 2020

a(n) = 2*a(n-1) + (2n-3)*a(n-2) - (2n-5)*a(n-3) + 2*a(n-4) - a(n-5).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + a(n-3) + a(n-4) + 2*sum( k=0..n-4, a(k) ).
a(n) ~ 2^(-1/2) * e^(-5/8) * (2n/e)^(n/2) * e^( sqrt(n/2) )  (conjectured).
a(n)/a(n-1) ~ sqrt(2n)  (conjectured).
a(n)/A047974(n) ~ 1/sqrt(e)  (conjectured).

More terms from Michel Marcus, Jul 28 2020

A271217 Number of symmetric reduced rearrangement maps.

Original entry on oeis.org

1, 2, 2, 6, 22, 50, 274, 598, 4486, 9570, 90914, 191398, 2201078, 4593554, 62012978, 128619510, 1993602406, 4115824322, 72026925634, 148169675590, 2889308674006

Offset: 0

Jonathan Burns, Apr 13 2016

a(n) is the number of reduced rearrangement maps on n blocks. A rearrangement map is a signed permutation, e.g., +2 -1 -3. If the permutation contains (i)(i+1) or -(i+1)-(i) for any i, then it is not reduced. The map a is symmetric if a=a^(AI) and a^A = a^I where A and I are the rotation involutions.

			For n=0 the a(0)=1 solution is { ∅ }
For n=1 the a(1)=2 solutions are { +1, -1 }
For n=2 the a(2)=2 solutions are { +2+1, -1-2 }
For n=3 the a(3)=6 solutions are { +3-2+1, -1+2-3, +3+2+1, -1-2-3, +1-2+3, -3+2-1 }

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

A271217 / A271216 ~ e^(-1/4).

Cf. A271212, A271214

Table[Round[2^n*Exp[-1/4]*(1-(1+(-1)^n)/(4 n))*Floor[n/2]!],{n,1,20}]

a(n) = round( 2^n * e^(-1/4) * ( 1 - (1 + (-1)^n)/(4n) ) * floor(n/2)! )
a(2k+1) = 2*a(2k) + a(2k-1) and a(2k) = (2k-1)*a(2k-1)+(2k-2)*a(2k-3)
a(n) ~ e^(-1/4) * 2^n * floor(n/2)!.
Conjecture: (-2*n+9)*a(n) -4*a(n-1) +(2*n-3)*(2*n-7)*a(n-2) -4*a(n-3) +2*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Jan 04 2017

A271216 a(n) = 2^n floor(n/2)!

Original entry on oeis.org

1, 2, 4, 8, 32, 64, 384, 768, 6144, 12288, 122880, 245760, 2949120, 5898240, 82575360, 165150720, 2642411520, 5284823040, 95126814720, 190253629440, 3805072588800

Offset: 0

Jonathan Burns, Apr 13 2016

Number of symmetric rearrangement maps, i.e., rearrangement maps which satisfy a=a^(AI) and a^A = a^I.

			For n=0 the a(0)=1 solution is { ∅ }
For n=1 the a(1)=2 solutions are { +1, -1 }
For n=2 the a(2)=4 solutions are { +1+2, -2-1, +2+1, -1-2 }
For n=3 the a(3)=8 solutions are { +1+2+3, -3-2-1, +3-2+1, -1+2-3, +3+2+1, -1-2-3, +1-2+3, -3+2-1 }

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

Cf. A000165, A271213.

Mathematica
```
Table[2^n*Floor[n/2]!,{n,0,20}]
```

a(n) = 2^n floor(n/2)!

A271215 Number of loop-free assembly graphs with n rigid vertices.

Original entry on oeis.org

1, 0, 1, 4, 24, 184, 1911, 24252, 362199, 6162080, 117342912, 2469791336, 56919388745, 1425435420600, 38543562608825, 1119188034056244, 34733368101580440, 1147320305439301344, 40190943859500501151, 1488212241729974297796, 58080468361734193793551

Offset: 0

Jonathan Burns, Apr 13 2016

nonn

Number of chord diagrams (equivalent up to reflection) that do not contain any simple chords, e.g., 121332 contains the simple chord 33.

			For n=0 the a(0)=1 solution is { ∅ }.
For n=1, a(1)=0 since the only assembly graph with one rigid vertex is the loop 11.
For n=2, the a(2)=1 solution is { 1212 }.
For n=3, the a(3)=4 solutions are { 121323, 123123, 123231, 123132 }.

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

Kristin DeSplinter, Satyan L. Devadoss, Jordan Readyhough, and Bryce Wimberly, Unfolding cubes: nets, packings, partitions, chords, arXiv:2007.13266 [math.CO], 2020. See Table 1 p. 15.

Cf. A000806, A132101, A271218.

(Table[Sum[Binomial[n,i]*(2*n-i)!/2^(n-i)*(-1)^(i)/n!,{i,0,n}],{n,0,20}]+RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,0,20}])/2

f(n) = sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) ); \\ A000806
lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); vector(nn-1, n, (va[n] + abs(f(n-1)))/2);} \\ Michel Marcus, Jul 28 2020

a(n) ~ (2n/e)^n / (e * sqrt(2)).
a(n) = (|A000806(n)| + A271218(n)) / 2.
a(n)/A132101(n) ~ 1/e.

A271214 Number of reduced rearrangement patterns with n blocks.

Original entry on oeis.org

1, 1, 2, 10, 71, 653, 7638, 104958, 1664083, 29740057, 591645738, 12959409010, 309898317151, 8032551265957, 224316415082750, 6714021923017318, 214415538303362411, 7277133405318569009, 261560966377901961810, 9925178291099012783322, 396498148141095399675511

Offset: 0

Jonathan Burns, Apr 13 2016

a(n) is the number of reduced rearrangement patterns, i.e., the number of reduced rearrangement map equivalence classes formed from the two rotation involutions.

			For n=0 the a(0)=1 solution is { ∅ }
For n=1 the a(1)=1 solution is { +1 }
For n=2 the a(2)=2 solutions are { +2+1, +1-2 }
For n=3 the a(3)=10 solutions are { +3-2+1, +1+3-2, +2-3+1, +1+3+2, +2+1-3, +3+1-2, +1-3+2, +3+2+1, +3+2-1, +1-2+3 }

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

G. C. Greubel, Table of n, a(n) for n = 0..400
J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

Cf. A271212, A271217.

Table[(Round[2^n*Exp[-1/2]*(n + 1/2)*(n - 1)!] + Round[2^n*Exp[ -1/4]*(1 - (1 + (-1)^n)/(4 n))*Floor[n/2]!])/4, {n,  1, 20}]

a(n) = ( round( 2^n e^(-1/2) (n+1/2) (n-1)! ) + round( 2^n e^(-1/4) (1-(1+(-1)^n)/4n)) floor(n/2)! ) / 4.
a(n) ~ sqrt( Pi*n / 8*e) * (2n / e)^n.
a(n) = (A271212(n) + A271217(n)) / 4.

A271213 a(n) = 2^(n-2) * (n! + floor(n/2)!).

Original entry on oeis.org

1, 1, 3, 14, 104, 976, 11616, 161472, 2582016, 46451712, 929003520, 20437463040, 490498375680, 12752940072960, 357082301399040, 10712468463943680, 342798990185594880, 11655165645170933760, 419585963202371911680, 15944266600833991311360, 637770664032408384307200

Offset: 0

Jonathan Burns, Apr 02 2016

a(n) is the number of rearrangement patterns, i.e., the number of rearrangement map equivalence classes.

			For n=1 the a(1)=1 solution is the equivalence class  {+1,-1}.For n=2 the a(2)=3 solutions are the equivalence classes {+1+2, -2-1}, {+1-2, +2-1, -2+1, -1+2}, and {+2+1, -1-2}

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

Partition of A000165 into equivalence classes.

Cf. A271214, A271216, A271217.

Mathematica
```
Table[2^(n-2)*(n!+Floor[n/2]!),{n,10}]
```

a(n)=2^(n-2)*(n!+floor(n/2)!)

a(n)~(pi*n/8)^(1/2) (2n/e)^n

A271212 a(n) = (2n-1)a(n-1) + 2(n-2)*a(n-2).

Original entry on oeis.org

1, 2, 6, 34, 262, 2562, 30278, 419234, 6651846, 118950658, 2366492038, 51837444642, 1239591067526, 32130200470274, 897265598318022, 26856087563449762, 857662151219847238, 29108533617158451714, 1046243865439580921606, 39700713164247881457698, 1585992592561492290028038

Offset: 0

Jonathan Burns, Apr 02 2016

a(n) is the number of reduced rearrangement maps on n blocks. A rearrangement map is a signed permutation, e.g., +2 -1 -3. If the permutation contains (i)(i+1) or -(i+1)-(i) for any i, then it is not reduced.

Number of permutations p of [2n] such that each element in p has exactly one neighbor whose value is smaller or larger by one. a(2) = 6: 1243, 2134, 2143, 3412, 3421, 4312. - Alois P. Heinz, May 24 2023

			For n=1 the a(1)=2 solutions are {+1,-1}.
For n=2 the a(2)=6 solutions are {+1-2,-1+2,-1-2,+2+1,+2-1,-2+1}. Note that {+1+2,-2-1} are not reduced rearrangement maps.

J. Burns, Counting a Class of Signed Permutations and Rigid Vertex Graphs related to Patterns of DNA Rearrangement, Preprint.

Alois P. Heinz, Table of n, a(n) for n = 0..404

Cf. A000165, A000354.

RecurrenceTable[{a[n]==(2n-1)*a[n-1]+2(n-2)*a[n-2],a[0]==1,a[1]==2},a[n],{n,0,10}]
Table[Round[Exp[-1/2]*(2n+1)*2^(n-1)*(n-1)!],{n,10}]

a(n) = (2n-1)*a(n-1) + 2*(n-2)*a(n-2); a(0)=1; a(1)=2;
a(n) = e^(-1/2)*(2n+1)*Gamma(n,-1/2)+(-1)^n
a(n) = e^(-1/2)*(2n+1)*2^(n-1)*(n-1)! + (-1)^(n+1)*(2n^2 + 3n)^(-1)* 2_F_2(1, n+1/2; n+1, n+5/2; -1/2)
a(n) = round( e^(-1/2)*(2n+1)*2^(n-1)*(n-1)! )
a(n) ~ (Pi*2n/e)^(1/2) * (2n/e)^n
From Peter Bala, May 29 2022: (Start)
a(n) = Sum_{k = 0..n-1} (-1)^(n-1+k)*2^(k+1)*(k+1)!*binomial(n-1,k) for n >= 1.
2*exp(-x)/(1 - 2*x)^2 = 2 + 6*x + 34*x^2/2! + 262*x^3/3! + 2562*x^4/4! + ... = Sum_{n >= 0} a(n+1)*x^n/n! is an e.g.f. for the sequence (a(n+1))n>=0.
a(n) = A000354(n) + A000354(n-1)  for n >= 1. (End)

cp's OEIS Frontend

User: Jonathan Burns

A271218 Number of symmetric linked diagrams with n links and no simple link.

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A271217 Number of symmetric reduced rearrangement maps.

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A271216 a(n) = 2^n floor(n/2)!

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A271215 Number of loop-free assembly graphs with n rigid vertices.

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A271214 Number of reduced rearrangement patterns with n blocks.

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A271213 a(n) = 2^(n-2) * (n! + floor(n/2)!).

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A271212 a(n) = (2n-1)*a(n-1) + 2*(n-2)*a(n-2).

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A271212 a(n) = (2n-1)a(n-1) + 2(n-2)*a(n-2).