A006385 -id:A006385 - cp's OEIS frontend

A006384 Number of sensed planar maps with n edges.

1, 2, 4, 14, 57, 312, 2071, 15030, 117735, 967850, 8268816, 72833730, 658049140, 6074058060, 57106433817, 545532037612, 5284835906037, 51833908183164, 514019531037910, 5147924676612282, 52017438279806634, 529867070532745464

Offset: 0

N. J. A. Sloane

The planar maps considered are connected and may contain loops and parallel edges. - Andrew Howroyd, Jan 13 2025

V. A. Liskovets, A census of nonisomorphic planar maps, in Algebraic Methods in Graph Theory, Vol. II, ed. L. Lovasz and V. T. Sos, North-Holland, 1981.
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Valery A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.
Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From _N. J. A. Sloane_, Jun 03 2012
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
N. J. A. Sloane, Notes
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
T. R. S. Walsh, Data (Preprint 1, Part 1)
T. R. S. Walsh, Data (Preprint 1, Part 2)
T. R. S. Walsh, Data (Preprint 1, Part 3)
T. R. S. Walsh, Notes
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
T. R. S. Walsh & N. J. A. Sloane, Correspondence, 1991
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.
Nicholas C. Wormald, On the number of planar maps, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)

Antidiagonal sums of A379430.

Cf. A000168 (rooted), A006385 (unsensed), A006443 (achiral), A006402 (2-connected).

with(numtheory): a:= n-> `if` (n=0, 1, floor (2*3^n /(n+1)/(n+2) *binomial(2*n, n) +add (phi(n/t) *3^t *binomial(2*t, t), t=divisors(n) minus {n}))/2/n +`if` (irem(n,2)=1, 2*3^((n-1)/2) /(n+1) *binomial(n-1, (n-1)/2), 2*(n-1) *3^((n-2)/2) /n/(n+2) *binomial(n-2, (n-2)/2))): seq (a(n), n=0..30); # Alois P. Heinz, Apr 24 2009

a[0] = 1; a[n_] := (1/(2n))*(2*(3^n/((n+1)*(n+2)))*Binomial[2n, n] + Sum[ EulerPhi[n/k]*3^k*Binomial[ 2k, k], {k, Most[ Divisors[n]]}]) + q[n]; q[n_?OddQ] := 2*(3^((n-1)/2)/(n+1))*Binomial[ n-1, (n-1)/2]; q[n_?EvenQ] := 2*(n-1)*(3^((n-2)/2)/(n*(n+2)))*Binomial[ n-2, (n-2)/2]; Table[ a[n], {n, 0, 21}] (* Jean-François Alcover, after Valery A. Liskovets *)

For n>0, a(n) = (1/2n)[A'(n)+sum_{kA000010, q(n)=(n+3) A'(n-1/2)/4 if n is odd and q(n) = (n-1)A'(n-2/2)/4 if n is even, where A'(n)=A000168(n), the number of rooted maps. - Valery A. Liskovets, May 27 2006
Equivalently, a(n) = (1/2n)[2*3^n/((n+1)(n+2))*binomial(2n,n) +sum_{kValery A. Liskovets, May 27 2006
a(n) ~ 12^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Sep 12 2014

More terms from Alois P. Heinz, Apr 24 2009

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).

Original entry on oeis.org

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.

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

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

A214814 Number of unsensed genus 2 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 4, 76, 1395, 24950, 427336, 6987100, 109761827, 1668376886, 24689351504, 357467967214, 5083309341304, 71209097157108, 984963603696282, 13477371260785608, 182698708325667710, 2456600457435363198, 32796863046711248526

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=2 of A379439.
Cf. A006301 (rooted), A104595 (sensed).
Cf. A006385, A006387, A214815, A214816.

More terms from Evgeniy Krasko, Sep 17 2017

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

A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 13, 20, 13, 3, 6, 35, 83, 83, 35, 6, 12, 104, 340, 504, 340, 104, 12, 27, 315, 1401, 2843, 2843, 1401, 315, 27, 65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65, 175, 3407, 24299, 82546, 149007, 149007, 82546, 24299, 3407, 175

Offset: 1

N. J. A. Sloane, Nov 07 2016

A(n,k) is also the number of multiquadrangulations of the sphere with n stable equilibria and k unstable equilibria.
From Andrew Howroyd, Jan 13 2025: (Start)
The planar maps considered are connected and may contain loops and parallel edges.
The number of edges is n + k - 2. (End)

			The array begins:
   1,    1,    1,     2,     3,     6,   12,   27, 65, ...
   1,    2,    5,    13,    35,   104,  315, 1021, ...
   1,    5,   20,    83,   340,  1401, 5809, ...
   2,   13,   83,   504,  2843, 15578, ...
   3,   35,  340,  2843, 21420, ...
   6,  104, 1401, 15578, ...
  12,  315, 5809, ...
  27, 1021, ...
  65, ...
  ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,    1;
   1,    2,    1;
   2,    5,    5,     2;
   3,   13,   20,    13,     3;
   6,   35,   83,    83,    35,    6;
  12,  104,  340,   504,   340,   104,   12;
  27,  315, 1401,  2843,  2843,  1401,  315,   27;
  65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65;
  ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, chapter 5.

Richard Kapolnai, Gabor Domokos, and Timea Szabo, Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also arXiv:1206.1698 [cs.DM], 2012. See Table 1.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 11-20.
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Antidiagonal sums are A006385.
Rows 1..2 (equally, columns 1..2) are A006082, A380239.
Cf. A269920 (rooted), A379430 (sensed), A379431 (achiral), A379432 (2-connected), A384963 (simple).

A(n,k) = A(k,n).

A(n,k) = (A379430(n,k) + A379431(n,k))/2. - Andrew Howroyd, Jan 14 2025

Missing terms inserted and definition edited by Andrew Howroyd, Jan 13 2025

A297880 Number of unsensed genus 4 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 7258, 688976, 37466297, 1512650776, 50355225387, 1461269893538, 38236656513725, 922552326544030, 20847359639841664, 446290728182323620, 9129236228868478458, 179639607187998993180, 3418366706444416598777

Offset: 0

Evgeniy Krasko, Jan 07 2018

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.

Column k=4 of A379439.
Cf. A215402 (rooted), A215019 (sensed).
Cf. A006385, A006387, A214814, A214815, A214816.

a(20) added by Andrew Howroyd, Jan 20 2025

A006443 Number of achiral planar maps with n edges.

Original entry on oeis.org

1, 2, 4, 14, 47, 184, 761, 3314, 14997, 69886, 333884, 1626998, 8067786, 40580084, 206734083, 1064666724, 5536480877, 29036188788, 153450351924, 816503772830, 4371551433888

Offset: 0

N. J. A. Sloane

An achiral map is a map with a sense-reversing automorphism.

The planar maps considered are connected and may contain loops and parallel edges. - Andrew Howroyd, Jan 13 2025

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

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 43.
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Column k=0 of A380234.

Cf. A006384 (sensed), A006385 (unsensed), A006444 (2-connected), A006445 (3-connected).

a(n) = 2*A006385(n) - A006384(n). [Liskovets eq 3a] - R. J. Mathar, Oct 01 2011

a(0)=1 prepended by Andrew Howroyd, Jan 13 2025

a(20) added by Andrew Howroyd, Jan 20 2025

A006389 Number of unsensed planar maps with n edges and without faces of degree 1.

Original entry on oeis.org

1, 1, 2, 6, 18, 68, 313, 1592, 9187, 57451, 384450, 2703970, 19769311

Offset: 0

N. J. A. Sloane

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

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.

Cf. A006385, A006388 (sensed), A379433 (rooted).

a(8)-a(12) from Sean A. Irvine, Mar 28 2017

a(0)=1 prepended by Andrew Howroyd, Jan 16 2025

cp's OEIS Frontend

A006384 Number of sensed planar maps with n edges.

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

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A006387 Number of unsensed genus 1 maps with n edges.

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A214814 Number of unsensed genus 2 maps with n edges.

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A214815 Number of unsensed genus 3 maps with n edges.

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A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

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A297880 Number of unsensed genus 4 maps with n edges.

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A006443 Number of achiral planar maps with n edges.

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