A170946 -id:A170946 - cp's OEIS frontend

A143660 Erroneous version of A170946.

2, 15, 20, 107

Offset: 1

dead

The theory of dessins d'enfants was initiated by A. Grothendieck. "In this work, using the matrix model approach... we listed all the dessins d'enfants with no more than 4 edges. There are two 1-edge dessins, both of them are of genus zero, fifteen 2-edge dessins, among them only one is of genus 1, twenty 3-edge dessins: 14 sperical [sic] and 6 of genus 1 and one hundred seven 4-edge dessins: 57 spherical dessins, 46 dessins of genus 1 and 4 dessins of genus 2. The total number of dessins is 134."

Note: The fifteen for n=2 is a typo, it should be 5. The sum: 2+5+20+107=134 adds up if we replace fifteen by 5, and moreover, the section on 2-edge dessins lists all 5 (not fifteen) dessins. The correct version of this sequence is given by A170946. - Mark van Hoeij, Jan 23 2011

N. M. Adrianov, N. Ya. Amburg, V. A. Dremov, Yu. A. Levitskaya, E. M. Kreines, Yu. Yu. Kochetkov, V. F. Nasretdinova and G. B. Shabat, Catalog of dessins d'enfants with <= 4 edges, arXiv:0710.2658 [math.AG], 2007.

A379438 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 2, 4, 1, 14, 6, 57, 46, 4, 312, 452, 106, 2071, 4852, 2382, 131, 15030, 52972, 46680, 8158, 117735, 587047, 830848, 313611, 14118, 967850, 6550808, 13804864, 9326858, 1369446, 8268816, 73483256, 218353000, 236095958, 74803564, 2976853, 72833730, 827801468, 3328822880, 5345316004, 3023693380, 391288854

Offset: 0

Andrew Howroyd, Jan 16 2025

			Triangle begins:
  n\k     [0]      [1]       [2]      [3]      [4]
  [0]      1;
  [1]      2;
  [2]      4,       1;
  [3]     14,       6;
  [4]     57,      46,        4;
  [5]    312,     452,      106;
  [6]   2071,    4852,     2382,     131;
  [7]  15030,   52972,    46680,    8158;
  [8] 117735,  587047,   830848,  313611,   14118;
  [9] 967850, 6550808, 13804864, 9326858, 1369446;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..120 (rows 0..20)
Antonio Breda d'Azevedo, Alexander Mednykh and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.
Timothy R. Walsh, Alain Giorgetti, and Alexander Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), no. 17, 2660--2671. MR2935417.

Row sums are A170946.
Columns 0..10 are A006384, A006386, A104595, A104596, A215019, A239918, A239919, A239921, A239922, A239923, A239924.
Cf. A269919 (rooted), A379439 (unsensed), A380234 (achiral), A380235.

A214816 Number of unsensed combinatorial maps with n edges on an orientable surface of any genus.

Original entry on oeis.org

1, 2, 5, 20, 96, 644, 5839, 67834, 970568, 16256556, 308620966, 6506035400, 150358570914, 3775903806928, 102348067516576, 2977979542305736, 92579723269733557, 3062602106878957610, 107418879166917701583, 3981908920500346885116, 155550644128029095714786

Offset: 0

N. J. A. Sloane, Jul 31 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..30
Antonio Breda d'Azevedo, Alexander Mednykh, and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps.
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Row sums of A379439.

Cf. A006385, A006387, A170946 (sensed), A170947 (achiral), A170948 (chiral pairs), A214814, A214815, A297880, A297881, A348798, A348800, A348801.

a(n) = (A170946(n) + A170947(n)) / 2. [Breda d'Azevedo, Mednykh & Nedela, Corollary 4.7] - Andrey Zabolotskiy, Jun 06 2024

a(12)-a(18) from Andrey Zabolotskiy, Jun 06 2024

a(19) onwards from Andrew Howroyd, Jan 27 2025

A170947 Number of achiral combinatorial maps with n edges.

Original entry on oeis.org

2, 5, 20, 85, 418, 2242, 12828, 77777, 493286, 3260485, 22314484, 157735801, 1147285362, 8570960234, 65611620808, 513963377327, 4113363020482, 33598074760393, 279764563749076, 2372822051513583, 20481425601917742, 179795508212739402, 1604084463778300348, 14536376462636666141

Offset: 1

N. J. A. Sloane, Feb 21 2010

nonn

Achiral maps are also called reflexible.

Antonio Breda d'Azevedo, Alexander Mednykh, and Roman Nedela, Table of n, a(n) for n = 1..30
Antonio Breda d'Azevedo, Alexander Mednykh, and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.

Row sums of A380234.

Cf. A170946 (sensed), A170948 (chiral pairs), A214816 (unsensed).

A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 18, 38, 34, 14, 3, 105, 275, 288, 154, 42, 6, 902, 2614, 3102, 1959, 705, 140, 14, 9749, 30346, 39242, 27898, 11956, 3142, 473, 34, 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95, 1915951, 6513999, 9484003, 7911844, 4230802, 1523176, 373188, 60614, 5969, 280

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of sensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |      1       2       3       4       5      6      7     8   9
----+----------------------------------------------------------------
  0 |      1
  1 |      1,      1
  2 |      2,      2,      1;
  3 |      5,      8,      5,      2;
  4 |     18,     38,     34,     14,      3;
  5 |    105,    275,    288,    154,     42,     6;
  6 |    902,   2614,   3102,   1959,    705,   140,    14;
  7 |   9749,  30346,  39242,  27898,  11956,  3142,   473,   34;
  8 | 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A170946.
Main diagonal is A002995(n+1).
Second diagonal gives A380237.
Columns 1..3 are A007769, A380618, A380619.
Cf. A053979 (rooted), A379430 (planar), A380616 (unsensed), A380617 (achiral).

InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k,r,y)={my(p=sumdiv(k,d,eulerphi(k/d)*y^d)/k); sum(i=0, r, abs(stirling(r,i,1))*p^i)/r!}
S(n,k,y)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)*C(k,r,y)), O(x*x^n))}
G(n,y='y)={prod(k=1, 2*n, S(n,k,y))}
T(n)={[Vecrev(p/y) | p<-Vec(y+InvEulerMTS(G(n)))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

A380626 Array read by antidiagonals: T(n,k) is the number of sensed k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 3, 1, 0, 1, 0, 7, 0, 1, 0, 1, 5, 29, 36, 11, 1, 0, 1, 0, 174, 0, 365, 0, 1, 0, 1, 18, 1475, 26614, 44106, 5250, 81, 1, 0, 1, 0, 16162, 0, 10107019, 0, 103801, 0, 1, 0, 1, 105, 214215, 102762168, 3703659517, 6605320523, 549530780, 2492164, 1228, 1, 0

Offset: 0

Andrew Howroyd, Jan 29 2025

The combinatorial maps considered are connected, unrooted, unlabeled, may have loops and parallel edges and are of any orientable genus.

			Array begins:
==================================================================
n\k | 1 2  3       4         5          6          7         8 ...
----+-------------------------------------------------------------
  0 | 1 1  1       1         1          1          1         1 ...
  1 | 0 1  0       2         0          5          0        18 ...
  2 | 1 1  3       7        29        174       1475     16162 ...
  3 | 0 1  0      36         0      26614          0 102762168 ...
  4 | 0 1 11     365     44106   10107019 3703659517 ...
  5 | 0 1  0    5250         0 6605320523 ...
  6 | 0 1 81  103801 549530780 ...
  7 | 0 1  0 2492164 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns 2..6 (odd columns with interspersed zeros) are A000012, A129114, A292206, A380627, A380628.
Row n=1 is A007769 (with interspersed zeros).
Cf. A170946, A380622 (rooted), A380629.

InvEulerT(v)={dirdiv(Vec(log(1+x*Ser(v)),-#v), vector(#v,n,1/n))}
D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}
G(n,m)={my(t=m*n); prod(k=1, t, my(A=O(x^(t\k+1)), p=serconvol(exp(A + D(m,k)), exp(A + D(2,k)))); sum(r=0, t\k, if(k*r%m==0, r!*polcoef(p,r)/(k^r)*x^(k*r/m)), O(x*x^n)) )}
T(n,k)=if(n==0, 1, InvEulerT(Vec(-1 + G(n,k), -n))[n])

A380629(n) = Sum_{d|2*n} T(d,2*n/d).

A170948 Number of chiral pairs of combinatorial maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 11, 226, 3597, 55006, 892791, 15763270, 305360481, 6483720916, 150200835113, 3774756521566, 102339496556342, 2977913930684928, 92579209306356230, 3062597993515937128, 107418845568842941190, 3981908640735783136040, 155550641755207044201203

Offset: 0

N. J. A. Sloane, Feb 21 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..30
Antonio Breda d'Azevedo, Alexander Mednykh, and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.

Cf. A170946 (sensed), A170947 (achiral), A214816 (unsensed).

a(n) = (A170946(n) - A170947(n)) / 2. [Breda d'Azevedo, Mednykh & Nedela, Corollary 4.8] - Andrey Zabolotskiy, Jun 06 2024

Name clarified, a(0)=0 prepended and a(19) onwards added by Andrew Howroyd, Jan 27 2025

A268558 Number of not necessarily connected sensed combinatorial maps with n edges.

Original entry on oeis.org

1, 2, 8, 34, 182, 1300, 12634, 153598, 2231004, 37250236, 699699968, 14574247086, 333121322514, 8286605836248, 222824153996898, 6439779836400464, 199051769194393718, 6552226226766384216, 228826838199807593530, 8450335361750379998822, 329002470731473098130572

Offset: 0

N. J. A. Sloane, Mar 02 2016

nonn

Original name: Arises in counting maps on a surface: see Coquereaux-Zuber (2015) for precise definition.

Number of nonisomorphic pairs (s,t) of permutations on a 2n-set where t is a fixed point free involution (i.e. all 2-cycles). Isomorphism is up to permutations of the n-set. - Andrew Howroyd, Jan 28 2025

Andrew Howroyd, Table of n, a(n) for n = 0..400 (terms 0..30 from Alois P. Heinz)
R. de Mello Koch, S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007. Different from a(7) onwards.
R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016),

Euler transform of A170946.

b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
S(n,k)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)), O(x*x^n))}
seq(n)={Vec(prod(k=1, 2*n, S(n,k)))} \\ Andrew Howroyd, Jan 28 2025

a(11)-a(18) from Euler transform of A170946 - R. J. Mathar, Apr 07 2022
a(0)=1 prepended and a(19)-a(20) (via A170946) from Alois P. Heinz, Jan 27 2025
Name edited by Andrew Howroyd, Jan 31 2025

A380365 Number of sensed combinatorial maps with n edges and without faces of degree 1.

Original entry on oeis.org

1, 1, 3, 11, 50, 365, 3782, 47935, 718202, 12245679, 233541489, 4920828395, 113495838798, 2843930973805, 76932818058660, 2234631397864123, 69368177318863458, 2291843543825994905, 80296746074069588380, 2973657775519950500203, 116065360915389313936460

Offset: 0

Andrew Howroyd, Jan 28 2025

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A006388 (planar), A170946, A380364 (rooted), A380366 (unsensed).

InvEulerT(v)={dirdiv(Vec(log(1+x*Ser(v)),-#v), vector(#v,n,1/n))}
b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k,r)={sum(i=0, r, (-1)^i/i!/k^i)}
S(n,k)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)*C(k,r)), O(x*x^n))}
seq(n)={concat([1], InvEulerT(Vec(-1 + prod(k=1, 2*n, S(n,k)))))}

A380629 Number of sensed regular combinatorial maps with n edges.

Original entry on oeis.org

1, 2, 3, 9, 26, 135, 1124, 11225, 143600, 2156862, 36069006, 681844857, 14387370477, 327462904319, 8171705457024, 221137571070305, 6373582250114091, 197210862517274355, 6521583445100185049, 227168823675390365225, 8396976723995537706278, 327880018217851412105973

Offset: 0

Andrew Howroyd, Jan 29 2025

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A170946, A380625 (rooted), A380626.

a(n)={if(n==0, 1, sumdiv(2*n, d, T(d,2*n/d)))} \\ T(n,k) defined in A380622.

a(n) = Sum_{d|2*n} A380626(d,2*n/d) for n > 0.

cp's OEIS Frontend

A143660 Erroneous version of A170946.

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A379438 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).

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A214816 Number of unsensed combinatorial maps with n edges on an orientable surface of any genus.

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A170947 Number of achiral combinatorial maps with n edges.

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A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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A380626 Array read by antidiagonals: T(n,k) is the number of sensed k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

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A170948 Number of chiral pairs of combinatorial maps with n edges.

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A268558 Number of not necessarily connected sensed combinatorial maps with n edges.

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A380365 Number of sensed combinatorial maps with n edges and without faces of degree 1.

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PARI

A380629 Number of sensed regular combinatorial maps with n edges.

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