A379438 -id:A379438 - cp's OEIS frontend

A379439 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).

1, 2, 4, 1, 14, 6, 52, 40, 4, 248, 320, 76, 1416, 2946, 1395, 82, 9172, 29364, 24950, 4348, 66366, 309558, 427336, 160050, 7258, 518868, 3365108, 6987100, 4696504, 688976, 4301350, 37246245, 109761827, 118353618, 37466297, 1491629, 37230364, 416751008, 1668376886, 2675297588, 1512650776, 195728778

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]     52,      40,       4;
  [5]    248,     320,      76;
  [6]   1416,    2946,    1395,      82;
  [7]   9172,   29364,   24950,    4348;
  [8]  66366,  309558,  427336,  160050,   7258;
  [9] 518868, 3365108, 6987100, 4696504, 688976;

Andrew Howroyd, Table of n, a(n) for n = 0..120 (rows 0..20)
Evgeniy Krasko and Alexander Omelchenko, Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus, arXiv:1712.10139 [math.CO], 2017, see Appendix Table 4.

Row sums are A214816.
Columns 0..8 are A006385, A006387, A214814, A214815, A297880, A297881, A348798, A348800, A348801.
Cf. A269919 (rooted), A379438 (sensed), A380234 (achiral), A380235.

T(n,k) = (A379438(n,k) + A380234(n,k))/2.

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

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.

A170946 Number of sensed unrooted combinatorial maps with n edges.

Original entry on oeis.org

1, 2, 5, 20, 107, 870, 9436, 122840, 1863359, 32019826, 613981447, 12989756316, 300559406027, 7550660328494, 204687564072918, 5955893472990664, 185158932576089787, 6125200100394894738, 214837724735760642773, 7963817561236130021156, 311101285883236139915989

Offset: 0

N. J. A. Sloane, Feb 21 2010

nonn

Also number of "dessins d'enfants" with n edges. - Mark van Hoeij, Jan 23 2011

a(n) also counts the Feynman diagrams of the QED vacuum polarization with 2*n vertices: fermion lines (resp. boson lines, vertices) of the Feynman diagrams correspond to the vertices (resp. edges, darts) of the combinatorial maps, and the circular order of the edges around each vertex in a map is encoded in the topology of the corresponding Feynman diagram. - Andrey Zabolotskiy, Jan 28 2025

Andrew Howroyd, Table of n, a(n) for n = 0..400 (terms 1..30 from Antonio Breda d'Azevedo, Alexander Mednykh and Roman Nedela)
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.
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.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section V. Computed up to a(5), plotted up to a(4).
R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: without large N, Phys Rev D, 85 (2012) 026007; arXiv:1110.4858 [hep-th], 2011-2012. The terms in Eq. (D.10) from a(7) on are erroneous.

Row sums of A379438 and A380615.
Cf. A170947 (achiral), A214816 (unsensed).
Cf. A268558 (inv. Euler Transf.)

a(0)=1 prepended by Andrew Howroyd, Jan 28 2025

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

A379439 Triangle read by rows: T(n,k) is the number of unsensed 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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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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A170946 Number of sensed unrooted combinatorial 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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