A069727 -id:A069727 - cp's OEIS frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

1, 3, 6, 20, 60, 291, 1310, 6975, 37746, 215602, 1262874, 7611156, 46814132, 293447817, 1868710728, 12068905911, 78913940784, 521709872895, 3483289035186, 23464708686960, 159346213738020, 1090073011199451, 7507285094455566, 52021636161126702

Offset: 1

Valery A. Liskovets, Dec 01 2003

a(n) is also the number of unrooted planar hypermaps with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets, Apr 13 2006

			The 3 Eulerian maps with 2 edges are the digon and two figure eight graphs ("8") in which both loops are colored, resp., black or white.

R. J. Mathar, Table of n, a(n) for n = 1..100
M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 2.
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.

Cf. A000257, A069727, A090372, A118094.

A090371 := proc(n)
    local s, d;
    if n=0 then
        1 ;
    else
        s := -2^n*binomial(2*n, n);
        for d in numtheory[divisors](n) do
            s := s+ numtheory[phi](n/d)*2^d*binomial(2*d, d)
        od;
        3/(2*n)*(2^n*binomial(2*n, n)/((n+1)*(n+2))+s/2);
    fi;
end proc:

h0[n_] := 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)); a[n_] := (h0[n] + DivisorSum[n, If[#>1, EulerPhi[#]*Binomial[n/#+2, 2]*h0[n/#], 0]&])/n; Array[a, 30] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2,2)*h0(n/d)))/n; \\ Michel Marcus, Dec 11 2014

More terms from Michel Marcus, Dec 11 2014

A069724 Number of nonisomorphic unrooted unicursal planar maps with n edges (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 2, 9, 38, 214, 1253, 7925, 51620, 346307, 2365886, 16421359, 115384738, 819276830, 5868540399, 42357643916, 307753571520, 2249048959624, 16520782751969, 121915128678131, 903391034923548, 6719098772562182

Offset: 1

Valery A. Liskovets, Apr 07 2002

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069720, A069727, A005470.

a[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(#-2) Binomial[2 #, #], 0]&] + If[OddQ[n], 2^((n-3)/2) Binomial[n-1, (n-1)/2], 2^((n-6)/2) Binomial[n, n/2]]; Array[a, 21] (* Jean-François Alcover, Sep 18 2016 *)

There is an easy formula.

a(n) ~ 8^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 28 2019

A103939 Number of unrooted Eulerian n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 3, 8, 32, 136, 722, 3924, 22954, 138316, 860364, 5472444, 35503288, 234070648, 1564945158, 10589356592, 72412611194, 499788291616, 3478059566250, 24383023246284, 172074483068320, 1221654305104920, 8720583728414354, 62560709120463028, 450854177292364660

Offset: 0

Valery A. Liskovets, Mar 17 2005

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A052701, A103940, A069727.

a[n_] := (1/(2n)) (2^n Binomial[2n, n]/(n+1) + Sum[Boole[0Jean-François Alcover, Aug 28 2019 *)

a(n)={if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

For n > 0, a(n) = (1/(2n))*(2^n*binomial(2n, n)/(n+1) + Sum_{0A000010.

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 29 2021

A103940 Number of unrooted bipartite n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 2, 5, 18, 72, 368, 1982, 11514, 69270, 430384, 2736894, 17752884, 117039548, 782480424, 5294705752, 36206357114, 249894328848, 1739030128872, 12191512867814, 86037243899240, 610827161152012, 4360291880624504, 31280354620428378, 225427088761560916, 1631398499577667252

Offset: 0

Valery A. Liskovets, Mar 17 2005

Bipartite planar maps are dual to Eulerian planar maps.

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Cf. A003645, A103939, A069727.

a[n_] := (1/(2 n)) (2^(n - 1) Binomial[2 n, n]/(n+1) + Sum[Boole[0 < k < n] EulerPhi[n/k] d[n/k] 2^(k-1) Binomial[2k, k], {k, Divisors[n]}]) + q[n];
d[n_] := If[EvenQ[n], 2, 1];
q[n_] := If[EvenQ[n], 0, 2^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
Array[a, 25] (* Jean-François Alcover, Aug 30 2019 *)

a(n)={if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

For n > 0, a(n) = (1/(2n))*[2^(n-1)*binomial(2n, n)/(n+1) + Sum_{0A000010, d(n)=2, q(n)=0 if n is even and d(n)=1, q(n)=2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd.

More terms from Jean-François Alcover, Aug 30 2019

a(0)=1 prepended by Andrew Howroyd, Mar 29 2021

A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 1, 3, 11, 62, 342, 2152, 13768, 91800, 622616, 4301792, 30100448, 213019072, 1521473984, 10954616064, 79420280064, 579300888960, 4248201302400, 31302536066560, 231638727063040

Offset: 1

Valery A. Liskovets, Apr 07 2002

There is an easy formula.

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069727, A003645, A069732.

a[1] = a[2] = 1;
a[n_] := With[{m = Floor[(n+1)/2]}, 1/n 2^(n-3) Binomial[2n-2, n-1] + 2^(m-3) Binomial[2m-2, m-1]];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 40, 93, 224, 538, 1344, 3352, 8448, 21573, 54912, 143037, 366080, 968083, 2489344, 6664856, 17199104, 46515759, 120393728, 328382874, 852017152, 2340706462, 6085836800, 16822999572, 43818024960, 121777594508, 317680680960, 887053276477

Offset: 1

Valery A. Liskovets, Dec 01 2003

M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
V. Liskovets, Some easily derivable integer sequences, J. Integer Seq., v.3 (2000), Article 00.2.2.
V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Cf. A052701, A069727, A090371.

A069727[n_] := (1/(2n)) (3*2^(n - 1) Binomial[2 n, n]/((n + 1) (n + 2)) + Sum[EulerPhi[n/k] d[n/k] 2^(k - 2) Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
q[n_?EvenQ] := 2^((n - 4)/2) Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2) Binomial[(n - 1), (n - 1)/2]/(n + 1);
d[n_] := 4 - Mod[n, 2];
h0[n_] := 3*2^(n - 1) Binomial[2n, n]/((n + 1)(n + 2));
A090371[n_] := (h0[n] + DivisorSum[n, If[# > 1, EulerPhi[#]*Binomial[n/# + 2, 2] h0[n/#], 0] &])/n;
a[n_] := 2 A069727[n] - A090371[n];
Array[a, 32] (* Jean-François Alcover, Aug 28 2019 *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a090371(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n;
d(n) = if (n%2, 3, 4);
q(n) = if (n%2, 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1), 2^((n-4)/2)*binomial(n, n/2)/(n+2));
a069727(n) = if (n==0, 1, q(n) + (3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)) + sumdiv(n, k, (k!=n)*eulerphi(n/k)*d(n/k)*2^(k-2)*binomial(2*k, k)))/(2*n));
a(n) = 2*a069727(n) - a090371(n); \\ Michel Marcus, Dec 11 2014

a(n) = 2*A069727(n) - A090371(n).

a(2k+1) = 2^k*Catalan(k) = A052701(k+1).

More terms from Michel Marcus, Dec 11 2014

A069730 Number of nonisomorphic unrooted unicursal planar maps with n edges.

Original entry on oeis.org

1, 2, 4, 13, 50, 248, 1407, 8600, 55154, 365292, 2473956, 17053468, 119191992, 842688120, 6015275094, 43292026736, 313788095994, 2288506113056, 16781638172458, 123656774440396, 915123392599456

Offset: 0

Valery A. Liskovets, Apr 07 2002

Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is, maps possessing an Eulerian path).

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069727, A069724.

A069724[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(# - 2) Binomial[2 #, #], 0] &] + If[OddQ[n], 2^((n - 3)/2) Binomial[n - 1, (n - 1)/2], 2^((n - 6)/2) Binomial[n, n/2]];
A069727[n_] := (1/(2 n))*(3*2^(n - 1)*Binomial[2 n, n]/((n + 1)*(n + 2)) + Sum[EulerPhi[n/k]*d[n/k]*2^(k - 2)*Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
q[n_?EvenQ] := 2^((n - 4)/2)*Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2)*Binomial[(n - 1), (n - 1)/2]/(n + 1);
d[n_] := 4 - Mod[n, 2];
a[n_] := A069727[n] + A069724[n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = A069727(n) + A069724(n).

cp's OEIS Frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

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A069724 Number of nonisomorphic unrooted unicursal planar maps with n edges (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A103939 Number of unrooted Eulerian n-edge maps in the plane (planar with a distinguished outside face).

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A103940 Number of unrooted bipartite n-edge maps in the plane (planar with a distinguished outside face).

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A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

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A069730 Number of nonisomorphic unrooted unicursal planar maps with n edges.

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