A006443 -id:A006443 - cp's OEIS frontend

A054936 Duplicate of A006443.

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

Offset: 1

A006385 Number of unsensed planar maps with n edges.

Original entry on oeis.org

1, 2, 4, 14, 52, 248, 1416, 9172, 66366, 518868, 4301350, 37230364, 333058463, 3057319072, 28656583950, 273298352168, 2645186193457, 25931472185976, 257086490694917, 2574370590192556, 26010904915620261

Offset: 0

N. J. A. Sloane

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).
T. R. S. Walsh, personal communication.

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, 2012. See Table 2.
Valery. A. Liskovets, A reductive technique for enumerating nonisomorphic planar maps, Discr. Math., v.156 (1996), 197-217.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Antidiagonal sums of A277741.
Column k=0 of A379439.
Cf. A000168 (rooted), A006384 (sensed), A006443 (achiral), A006403 (2-connected), A090376.
Cf. A006387 (genus 1), A214814 (genus 2), A214815 (genus 3), A214816.

a(n) = (A006384(n) + A006443(n))/2. - Andrew Howroyd, Jan 13 2025

a(18)-a(19) added by Andrew Howroyd, Jan 13 2025

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

A006384 Number of sensed planar maps with n edges.

Original entry on oeis.org

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

A380234 Triangle read by rows: T(n,k) is the number of achiral combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 2, 4, 1, 14, 6, 47, 34, 4, 184, 188, 46, 761, 1040, 408, 33, 3314, 5756, 3220, 538, 14997, 32069, 23824, 6489, 398, 69886, 179408, 169336, 66150, 8506, 333884, 1009234, 1170654, 611278, 129030, 6405, 1626998, 5700548, 7930892, 5279172, 1608172, 168702, 8067786, 32341002, 52930196, 43429578, 17758601, 3080190, 128448

Offset: 0

Andrew Howroyd, Jan 17 2025

Achiral maps are also called reflexible.

			Triangle starts:
  n\k    [0]     [1]     [2]    [3]   [4]
  [0]     1;
  [1]     2;
  [2]     4,      1;
  [3]    14,      6;
  [4]    47,     34,      4;
  [5]   184,    188,     46;
  [6]   761,   1040,    408,    33;
  [7]  3314,   5756,   3220,   538;
  [8] 14997,  32069,  23824,  6489,  398;
  [9] 69886, 179408, 169336, 66150, 8506;
  ...

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.
Evgeniy Krasko, Reflexible maps with E edges on orientable surfaces of genus G, 2017 (data file).

Row sums are A170947.
Column 0 is A006443.
Cf. A379438 (sensed), A379439 (unsensed).

A379431 Array read by antidiagonals: A(n,k) is the number of achiral 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, 12, 17, 12, 3, 6, 28, 58, 58, 28, 6, 10, 68, 179, 247, 179, 68, 10, 20, 157, 538, 942, 942, 538, 157, 20, 35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35, 70, 845, 4288, 11424, 18316, 18316, 11424, 4288, 845, 70

Offset: 1

Andrew Howroyd, Jan 14 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			==================================================
n\k |  1   2    3     4     5     6     7    8 ...
----+---------------------------------------------
  1 |  1   1    1     2     3     6    10   20 ...
  2 |  1   2    5    12    28    68   157  372 ...
  3 |  1   5   17    58   179   538  1531 4288 ...
  4 |  2  12   58   247   942  3388 11424 ...
  5 |  3  28  179   942  4345 18316 ...
  6 |  6  68  538  3388 18316 ...
  7 | 10 157 1531 11424 ...
  8 | 20 372 4288 ...
  ...
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,  12,   17,   12,    3;
   6,  28,   58,   58,   28,    6;
  10,  68,  179,  247,  179,   68,   10;
  20, 157,  538,  942,  942,  538,  157,  20;
  35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35;
  ...

Antidiagonal sums are A006443.
Column 1 is A210736(n-1).
Cf. A269920 (rooted), A277741 (unsensed), A379430 (sensed).

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

A006444 Number of achiral 2-connected planar maps with n edges.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 30, 77, 196, 525, 1414, 3960, 11056, 31636, 90818, 264657, 774146, 2289787, 6798562, 20354005, 61164374, 184985060, 561433922, 1712696708, 5241637812

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 44.

Cf. A006402 (sensed), A006403 (unsensed), A006443 (connected), A006445 (3-connected).

A054937 Number of chiral pairs of planar maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 5, 64, 655, 5858, 51369, 448982, 3967466, 35603366, 324990677, 3016738988, 28449849867, 272233685444, 2639649712580, 25902435997188

Offset: 0

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A006384, A006385, A006443.

a(n) = A006385(n) - A006443(n). - R. J. Mathar, Oct 01 2011

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

cp's OEIS Frontend

A054936 Duplicate of A006443.

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A006385 Number of unsensed planar maps with n edges.

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A006384 Number of sensed planar maps with n edges.

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A380234 Triangle read by rows: T(n,k) is the number of achiral combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).

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A379431 Array read by antidiagonals: A(n,k) is the number of achiral planar maps with n vertices and k faces, n >= 1, k >= 1.

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A006444 Number of achiral 2-connected planar maps with n edges.

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A054937 Number of chiral pairs of planar maps with n edges.

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