A104596 -id:A104596 - cp's OEIS frontend

A104742 Number of rooted maps of (orientable) genus 3 containing n edges.

1485, 113256, 5008230, 167808024, 4721384790, 117593590752, 2675326679856, 56740864304592, 1137757854901806, 21789659909226960, 401602392805341924, 7165100439281414160, 124314235272290304540, 2105172926498512761984, 34899691847703927826500, 567797719808735191344672, 9084445205688065541367710

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)
E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Combin. Theory, B 53 (1991), 293-299.
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).

Column k=3 of A238396.
Cf. A104596 (unrooted maps).
Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, this sequence, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 3];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jul 20 2018 *)

A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
A104742_ser(N) = {
  my(y=A005159_ser(N+1));
  y*(y-1)^6*(460*y^8 - 3680*y^7 + 63055*y^6 - 198110*y^5 + 835954*y^4 - 1408808*y^3 + 1986832*y^2 - 1462400*y + 547552)/(81*(y-2)^12*(y+2)^7)
};
Vec(A104742_ser(17))  \\ Gheorghe Coserea, Jun 02 2017

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.

A006386 Number of sensed genus 1 maps with n edges.

Original entry on oeis.org

1, 6, 46, 452, 4852, 52972, 587047, 6550808, 73483256, 827801468, 9360123740, 106189359544, 1208328304864, 13787042250528, 157700137398689, 1807893066408464, 20768681225892328, 239037464947999900, 2755989928117365244, 31826208029615881656, 368074022535205870382

Offset: 2

N. J. A. Sloane

A genus 1 map can be called a toroidal map.

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

T. D. Noe, Table of n, a(n) for n = 2..30 (from Mednykh and Nedela)
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, J. Combin. Th. B, 96 (2006), 706-729.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol.502, pp.4-15, (September-2013).
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=1 of A379438.

Cf. A006300 (rooted), A006384 (planar), A006387 (unsensed), A104595, A104596, A215019.

More terms from Valery A. Liskovets, Mar 22 2005

Edited by N. J. A. Sloane, May 23 2008

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.

A214815 Number of unsensed genus 3 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 82, 4348, 160050, 4696504, 118353618, 2675297588, 55758114082, 1091344752470, 20318440463052, 363171011546210, 6275111078422480, 105369657960443204, 1726590417107274316, 27699670730854989616, 436246336648672487876

Offset: 0

N. J. A. Sloane, Jul 31 2012

nonn

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.
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Column k=3 of A379439.
Cf. A104742 (rooted), A104596 (sensed).
Cf. A006385, A006387, A214814, A214816.

More terms 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

A104742 Number of rooted maps of (orientable) genus 3 containing n edges.

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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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A006386 Number of sensed genus 1 maps with n edges.

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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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A214815 Number of unsensed genus 3 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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