A006386 -id:A006386 - cp's OEIS frontend

A006300 Number of rooted maps with n edges on torus.

1, 20, 307, 4280, 56914, 736568, 9370183, 117822512, 1469283166, 18210135416, 224636864830, 2760899996816, 33833099832484, 413610917006000, 5046403030066927, 61468359153954656, 747672504476150374, 9083423595292949240, 110239596847544663002, 1336700736225591436496, 16195256987701502444284

Offset: 2

N. J. A. Sloane

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. D. Noe, Table of n, a(n) for n = 2..100
D. Arquès, Relations fonctionnelles et dénombrement des cartes pointées sur le tore, J. Combin. Theory Ser. B, 43 (1987), 253-274.
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271.
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, Discrete Mathematics, Volume 310, Issue 3, 6 February 2010, pp. 518-526.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
T. R. S. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol. 502, pp. 4-15, (September-2013).

Column k=1 of A238396.
Cf. A007137, A006386.
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, this sequence, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

R:=sqrt(1-12*x): seq(coeff(convert(series((R-1)^2/(12*R^2*(R+2)),x,50),polynom),x,n),n=2..25); (Pab Ter)

Drop[With[{c=Sqrt[1-12x]},CoefficientList[Series[(c-1)^2/(12c^2 (c+2)), {x,0,30}],x]],2] (* Harvey P. Dale, Jun 14 2011 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A006300_ser(N) = my(y = A005159_ser(N+1)); y*(y-1)^2/(3*(y-2)^2*(y+2));
Vec(A006300_ser(21)) \\ Gheorghe Coserea, Jun 02 2017

G.f.: (R-1)^2/(12*R^2*(R+2)) where R=sqrt(1-12*x); a(n) is asymptotic to 12^n/24. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
a(n) = Sum_{k=0..n-2} 2^(n-3-k)*(3^(n-1)-3^k)*binomial(n+k,k). - Ruperto Corso, Dec 18 2011
D-finite with recurrence: n*a(n) +22*(-n+1)*a(n-1) +4*(22*n-65)*a(n-2) +96*(5*n-4)*a(n-3) +576*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Feb 20 2020

Bender et al. give 20 terms.
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
More terms from Joerg Arndt, Feb 26 2014

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.

A104596 Number of unrooted maps with n edges of (orientable) genus 3.

Original entry on oeis.org

131, 8158, 313611, 9326858, 236095958, 5345316004, 111472798586, 2182345314816, 40634231364914, 726322104184848, 12550075287918360, 210738250570954064, 3453173212810875280, 55399287587418128520, 872492296405529104608, 13518993329700676078500, 206464663769623968602698

Offset: 6

Valery A. Liskovets, Mar 22 2005

nonn

T. D. Noe, Table of n, a(n) for n=6..30 (from Mednykh and Nedela)
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
Timothy R. S. Walsh, Alain Giorgetti, 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. - From _N. J. A. Sloane_, Aug 01 2012

Column k=3 of A379438.
Cf. A104742 (rooted), A214815 (unsensed).
Cf. A006386, A104595 (genus 2), A215019 (genus 4).

A215019 Number of unrooted maps with n edges of (orientable) genus 4.

Original entry on oeis.org

14118, 1369446, 74803564, 3023693380, 100692692173, 2922359760376, 76471600288836, 1845089145736960, 41694584320696782, 892580319444417876, 18258463136626650660, 359279139700128276168, 6836732826365623258492, 126347598971804884131800, 2275643837092089686415858

Offset: 8

N. J. A. Sloane, Aug 01 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 8..45
Alexander Mednykh, Alain Giorgetti, Enumeration of genus four maps by number of edges. Ars Mathematica Contemporanea 4 (2011), 351-361. MR2842107
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
Timothy R. S. Walsh, Alain Giorgetti, 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

Column k=4 of A379438.
Cf. A215402 (rooted), A297880 (unsensed).
Cf. A006386, A104595, A104596 (genus 3), A239918 (genus 5), A239919 (genus 6).

a(12) onwards added by Andrew Howroyd, Jan 18 2025

A104595 Number of unrooted maps with n edges of (orientable) genus 2.

Original entry on oeis.org

4, 106, 2382, 46680, 830848, 13804864, 218353000, 3328822880, 49325772812, 714586880940, 10164338225482, 142403410942816, 1969831979334086, 26954132420126920, 365393525753591368, 4913176199287631232, 65593569635906036912, 870192550284377429780, 11479539192932030062066

Offset: 4

Valery A. Liskovets, Mar 22 2005

nonn

T. D. Noe, Table of n, a(n) for n=4..30 (from Mednykh and Nedela)
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
Timothy R. S. Walsh, Alain Giorgetti, 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. - From _N. J. A. Sloane_, Aug 01 2012

Column k=2 of A379438.
Cf. A006301 (rooted), A214814 (unsensed).
Cf. A006386, A104595, A104596, A215019.

A006387 Number of unsensed genus 1 maps with n edges.

Original entry on oeis.org

0, 0, 1, 6, 40, 320, 2946, 29364, 309558, 3365108, 37246245, 416751008, 4696232371, 53186743416, 604690121555, 6896534910612, 78867385697513, 904046279771682, 10384916465797240, 119522063788612992, 1378014272286250059

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Evgeniy Krasko and Alexander Omelchenko, Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus, arXiv:1712.10139 [math.CO], 2017.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Column k=1 of A379439.

Cf. A006300 (rooted), A006385 (planar), A006386 (sensed), A214814, A214815, A214816.

a(12)-a(20) from Evgeniy Krasko, Sep 17 2017

A239918 Number of unrooted maps with n edges of (orientable) genus 5.

Original entry on oeis.org

2976853, 391288854, 28036387466, 1449247494892, 60441165724160, 2158501051914340, 68463726004852884, 1976314846820429680, 52825750657523709792, 1324265997531577820388, 31439565426089264422698, 712298211293218414835136

Offset: 10

Alain Giorgetti, Mar 29 2014

Timothy R. S. Walsh, Alain Giorgetti, 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

Column k=5 of A379438.
Cf. A238355 (rooted), A297881 (unsensed).
Cf. A006386, A104595, A104596, A215019.

A239919 Number of unrooted maps with n edges of (orientable) genus 6.

Original entry on oeis.org

1013582110, 169857459866, 15202559941266, 964949198577434, 48695382495936280, 2077641325234343160, 77856954571873092792, 2629247239663140056192, 81498694882848919958250, 2350709723279734060249864, 63759822591543968176310334, 1639827731755812039740785472, 40257891518670970262942165090

Offset: 12

Alain Giorgetti, Mar 29 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 12..60
Timothy R. S. Walsh, Alain Giorgetti, 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.

Column k=6 of A379438
Cf. A238356 (rooted), A348798 (unsensed).
Cf. A006386, A104595, A104596, A215019, A239918.

a(22) onwards added by Andrew Howroyd, Jan 18 2025

A239921 Number of unrooted maps with n edges of (orientable) genus 7.

Original entry on oeis.org

508233789579, 104295987346126, 11269592389125547, 852994611088758224, 50777879440443305426, 2531246455428148382456, 109880399953287962099588, 4265557888300762164284822, 150940131172496245801920542, 4938911033961317567088755908, 151101665358744941452325232448

Offset: 14

Alain Giorgetti, Mar 29 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 14..60
Timothy R. S. Walsh, Alain Giorgetti, 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.

Column k=7 of A379438.
Cf. A238357 (rooted), A348800 (unsensed).
Cf. A006386, A104595, A104596, A215019, A239918, A239919.

a(23) onwards added by Andrew Howroyd, Jan 18 2025

A239922 Number of unrooted maps with n edges of (orientable) genus 8.

Original entry on oeis.org

352755124921122, 86116887841273186, 10954787876407932816, 967146341367928365308, 66599326875830666763353, 3811863659211606517416928, 188710867264106506704457217, 8303453286421604392505856232, 331175730212422476849562734689, 12151338155016475304016716988472

Offset: 16

Alain Giorgetti, Mar 29 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 16..60
Timothy R. S. Walsh, Alain Giorgetti, 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.

Column k=8 of A379438.
Cf. A238358 (rooted), A348801 (unsensed).
Cf. A006386, A104595, A104596, A215019, A239918, A239919, A239921.

a(24) onwards added by Andrew Howroyd, Jan 18 2025

cp's OEIS Frontend

A006300 Number of rooted maps with n edges on torus.

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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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A104596 Number of unrooted maps with n edges of (orientable) genus 3.

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A215019 Number of unrooted maps with n edges of (orientable) genus 4.

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A104595 Number of unrooted maps with n edges of (orientable) genus 2.

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A006387 Number of unsensed genus 1 maps with n edges.

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A239918 Number of unrooted maps with n edges of (orientable) genus 5.

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A239919 Number of unrooted maps with n edges of (orientable) genus 6.

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A239921 Number of unrooted maps with n edges of (orientable) genus 7.

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A239922 Number of unrooted maps with n edges of (orientable) genus 8.

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