A379430 -id:A379430 - 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

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

A380237 Number of sensed planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 14, 42, 140, 473, 1670, 5969, 21679, 79419, 293496, 1091006, 4078213, 15312150, 57721030, 218333832, 828408842, 3151769615, 12020870753, 45949957412, 176001205559, 675384194565, 2596119292840, 9994894356158, 38535398284100, 148772774499015, 575079507042663

Offset: 1

Andrew Howroyd, Jan 19 2025

nonn

Also, by duality the number of sensed planar maps with n faces and 2 vertices.

The number of edges is n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column 2 of A379430.
Cf. A000346 (rooted), A380238 (achiral), A380239 (unsensed), A060404 (with a distinguished face), A103943 (with a distinguished vertex).
Cf. A000108, A210736.

a(n) = {(binomial(n - 1, (n - 1)\2) + sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n)/2}

seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k))/2}

a(n) = (A210736(n) + A060404(n))/2.
a(n) = (1/(2*n))*(n*binomial(n-1, floor((n-1)/2)) + Sum_{d|n} phi(n/d)*(2^(2*d-1) - binomial(2*d-1, d))).
G.f.: (1/2)*(1/(1 - x*C(x^2)) - 1 - Sum_{k>=1} log(1 - C(x^k)) * phi(k)/k), where C(x) is the g.f. of A000108.

A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 18, 38, 34, 14, 3, 105, 275, 288, 154, 42, 6, 902, 2614, 3102, 1959, 705, 140, 14, 9749, 30346, 39242, 27898, 11956, 3142, 473, 34, 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95, 1915951, 6513999, 9484003, 7911844, 4230802, 1523176, 373188, 60614, 5969, 280

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of sensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |      1       2       3       4       5      6      7     8   9
----+----------------------------------------------------------------
  0 |      1
  1 |      1,      1
  2 |      2,      2,      1;
  3 |      5,      8,      5,      2;
  4 |     18,     38,     34,     14,      3;
  5 |    105,    275,    288,    154,     42,     6;
  6 |    902,   2614,   3102,   1959,    705,   140,    14;
  7 |   9749,  30346,  39242,  27898,  11956,  3142,   473,   34;
  8 | 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A170946.
Main diagonal is A002995(n+1).
Second diagonal gives A380237.
Columns 1..3 are A007769, A380618, A380619.
Cf. A053979 (rooted), A379430 (planar), A380616 (unsensed), A380617 (achiral).

InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k,r,y)={my(p=sumdiv(k,d,eulerphi(k/d)*y^d)/k); sum(i=0, r, abs(stirling(r,i,1))*p^i)/r!}
S(n,k,y)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)*C(k,r,y)), O(x*x^n))}
G(n,y='y)={prod(k=1, 2*n, S(n,k,y))}
T(n)={[Vecrev(p/y) | p<-Vec(y+InvEulerMTS(G(n)))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 8, 6, 2, 1, 6, 29, 60, 73, 52, 25, 6, 2, 14, 113, 388, 768, 903, 728, 379, 136, 26, 6, 34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17, 95, 1763, 12650, 49806, 123547, 210314, 255884, 228807, 150929, 73428, 25536, 6142, 892, 73

Offset: 1

Andrew Howroyd, Jun 13 2025

Equivalently, T(n,k) is the number of sensed simple planar maps with n vertices and k faces.
The number of edges is n+k-2.
Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 1..14 of this table.

			Triangle begins:
   1;
   1;
   1,   1,
   2,   2,    1,    1,
   3,   8,    8,    6,     2,     1,
   6,  29,   60,   73,    52,    25,     6,    2,
  14, 113,  388,  768,   903,   728,   379,  136,   26,   6,
  34, 444, 2303, 6584, 11782, 14321, 12113, 7298, 3048, 872, 147, 17;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..158 (rows 1..14)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 19-22.

Row sums are A384965.
Antidiagonal sums are A006394.
Columns 1..2 are A002995, A384966.
Cf. A379430 (not necessarily simple), A342059 (2-connected), A239893 (3-connected), A384963 (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).

A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 7, 22, 76, 271, 1001, 3765, 14381, 55450, 214880, 835663, 3255652, 12698352, 49559793, 193513944, 755852101, 2953214386, 11541989533, 45123241746, 176465152051, 690340349398, 2701579878022, 10576116931462, 41418132927403, 162259989848094, 635899817853002, 2492993368347594

Offset: 1

Andrew Howroyd, Jun 15 2025

nonn

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes.

Column 2 of A384963.
Also subdiagonal of A379430.
Cf. A001429, A006081 (cycle is loop), A380239 (not necessarily simple), A384966 (sensed version).

G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x^(n+2))))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1 - x*(g^2/(1 - x*g)^2 + g) - x^2*(g^4/(1 - x*g)^4 + 3*g^2)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2}
seq(n)={Vec(G1(n)+G2(n), -n)/4}

A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 12, 8, 2, 10, 48, 64, 25, 3, 26, 196, 412, 314, 78, 6, 80, 798, 2458, 2976, 1478, 270, 14, 246, 3248, 13452, 23588, 18844, 6748, 926, 34, 810, 13184, 70330, 166050, 192096, 110714, 30168, 3305, 95, 2704, 53416, 353716, 1074472, 1676668, 1397484, 613884, 132734, 11868, 280, 9252

Offset: 1

Andrew Howroyd, Jan 21 2025

The number of edges is n + k - 2.

			Array begins:
==============================================================
n\k |  1    2      3      4       5       6       7      8 ...
----+---------------------------------------------------------
  1 |  1    1      2      4      10      26      80    246 ...
  2 |  1    3     12     48     196     798    3248  13184 ...
  3 |  1    8     64    412    2458   13452   70330 353716 ...
  4 |  2   25    314   2976   23588  166050 1074472 ...
  5 |  3   78   1478  18844  192096 1676668 ...
  6 |  6  270   6748 110714 1397484 ...
  7 | 14  926  30168 613884 ...
  8 | 34 3305 132734 ...
   ...

Columns 1..2 are A002995, A060404.
Rows 1..2 are A003239(n-1), A103943.
Antidiagonal sums are A103937.
Cf. A269920 (rooted), A379430 (sensed with no root).

cp's OEIS Frontend

A006384 Number of sensed planar 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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A380237 Number of sensed planar maps with n vertices and 2 faces.

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A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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A384964 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of connected simple planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 1, k=1..max(1,2*n-4).

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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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A384967 Number of unsensed simple planar maps with n vertices and 2 faces.

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A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

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