A046653 -id:A046653 - cp's OEIS frontend

A000087 Number of unrooted nonseparable planar maps with n edges and a distinguished face.

2, 1, 2, 4, 10, 37, 138, 628, 2972, 14903, 76994, 409594, 2222628, 12281570, 68864086, 391120036, 2246122574, 13025721601, 76194378042, 449155863868, 2666126033850, 15925105028685, 95664343622234, 577651490729530

Offset: 1

N. J. A. Sloane

nonn

The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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=1..200
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
W. G. Brown, Enumeration of non-separable planar maps
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Row sums of A046653.

Cf. A006402, A103938.

q[n_] := If[EvenQ[n], 0, 2(n+1)Binomial[3(n+1)/2, (n+1)/2]/(3(3n-1)(3n+1)) ]; a[n_] := (1/(3n))((n+2)Binomial[3n, n]/((3n-2)(3n-1)) + Sum[EulerPhi[ n/k] Binomial[3k, k], {k, Divisors[n] // Most}]) + q[n]; Array[a, 30] (* Jean-François Alcover, Feb 04 2016, after Valery A. Liskovets *)

q(n) = if(n%2, 2*(n + 1)*binomial(3*(n + 1)/2, (n + 1)/2) / (3*(3*n - 1)*(3*n + 1)), 0);
a(n) = (1/(3*n)) * ((n + 2) * binomial(3*n, n)/((3*n - 2) * (3*n - 1)) + sum(k=1, n - 1, if(Mod(n, k)==0, eulerphi(n/k) * binomial(3*k, k)))) + q(n); \\ Indranil Ghosh, Apr 04 2017

a(n) = (1/3n)[(n+2)binomial(3n, n)/((3n-2)(3n-1)) + Sum_{0A000010, q(n)=0 if n is even and q(n)=2(n+1)binomial(3(n+1)/2, (n+1)/2)/(3(3n-1)(3n+1)) if n is odd. - Valery A. Liskovets, Mar 17 2005

a(n) ~ 3/(8 * sqrt(3*Pi))*(27/4)^n / n^(5/2). - Cedric Lorand, Apr 18 2022

More terms from T. D. Noe, Mar 14 2007
Name corrected by Cyril Banderier, Apr 04 2017
Name clarified by Andrew Howroyd, Mar 29 2021

A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2n+1 edges and a fixed outer face of 2k edges which are invariant under a rotation of a 1/2 turn.

Original entry on oeis.org

1, 2, 2, 7, 8, 3, 30, 34, 21, 4, 143, 160, 114, 44, 5, 728, 806, 609, 308, 80, 6, 3876, 4256, 3315, 1908, 715, 132, 7, 21318, 23256, 18444, 11420, 5185, 1482, 203, 8, 120175, 130416, 104652, 67856, 34520, 12600, 2814, 296, 9, 690690, 746350, 603801, 404016, 221300, 93924, 27965, 4984, 414, 10

Offset: 1

Emeric Deutsch, Mar 03 2004

Table II in the Brown reference.

			Triangle begins:
    1;
    2,   2;
    7,   8,   3;
   30,  34,  21,  4;
  143, 160, 114, 44, 5;
  ...
The T(n,n) = n solutions correspond to a regular polygon with 2n vertices and a single diagonal joining two diametrically opposite vertices. - _Andrew Howroyd_, Mar 29 2021

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Column 1 gives A006013, column 2 gives A046649, row sums give A000305.
Same as A046652 but with rows reversed.
Cf. A046653, A091599.

T := proc(n,k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!,j=k..min(n,2*k-1))/(2*n-k+1)! else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Maple *)

T(n,k) = {k*sum(j=k, min(n, 2*k-1), (2*j-k+1)*(j-1)!*(3*n-k-j)!/((j-k+1)!*(j-k)!*(2*k-j-1)!*(n-j)!))/(2*n-k+1)!}
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

T(n, k) = k*(Sum_{j=k..min(n, 2*k-1)} (2*j-k+1)*(j-1)!*(3*n-k-j)!/((j-k+1)!*(j-k)!*(2*k-j-1)!*(n-j)!))/(2*n-k+1)! for k<=n and T(n, k)=0 for k>n.

Name clarified by Andrew Howroyd, Mar 29 2021

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with rn edges and a fixed outer face of rk edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 26, 12, 1, 110, 120, 75, 20, 1, 546, 594, 416, 174, 30, 1, 2856, 3094, 2289, 1176, 350, 42, 1, 15504, 16728, 12768, 7322, 2880, 636, 56, 1, 86526, 93024, 72420, 44388, 20475, 6324, 1071, 72, 1, 493350, 528770, 417240, 267240, 136252, 51495, 12740, 1700, 90, 1

Offset: 1

Emeric Deutsch, Mar 03 2004

Table I in the Brown reference.

			Triangle starts:
    1;
    2,   1;
    6,   6,  1;
   24,  26, 12,  1;
  110, 120, 75, 20, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Column 1 gives A046646, column 2 gives A046647, row sums give A000259.
Same as A046651 but with rows reversed.
Cf. A046653, A091665.

T := proc(n,k) if k<=n then k*sum((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!,j=k..min(n,2*k))/(2*n-k)! else 0 fi end: seq(seq(T(n,k), k=1..n),n=1..11);

T(n, k) = k*sum(j=k, min(n, 2*k), (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

T(n, k) = k*(Sum_{j=k..min(n, 2*k)} (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!

Name clarified by Andrew Howroyd, Mar 29 2021

A046650 Number of sensed nonseparable (2-connected) planar maps with n edges and a distinguished face of size 2.

Original entry on oeis.org

1, 1, 2, 4, 14, 49, 216, 984, 4862, 24739, 130338, 701584, 3852744, 21489836, 121525520, 695307888, 4019381790, 23446201495, 137875564710, 816646459860, 4868578092510, 29196022525905, 176022392938080, 1066433501134560, 6490009570139784, 39659537885087124, 243278423033093336, 1497584057249141728, 9249144367260811824

Offset: 2

N. J. A. Sloane

From R. J. Mathar, Apr 13 2019: (Start)
Table III with row sums A000087 is (A046653 row-reversed):
     1;
     1,    1;
     2,    1,    1;
     4,    3,    2,    1;
    14,   12,    8,    2,   1;
    49,   43,   30,   12,   3,   1;
   216,  189,  134,   63,  22,   3,  1;
   984,  888,  608,  323, 133,  31,  4, 1;
  4862, 4332, 2988, 1671, 759, 238, 48, 4, 1;
  ...
(End)
Equivalently, by duality, a(n) is the number of sensed nonseparable planar maps with n edges and a distinguished vertex of degree 2.  - Andrew Howroyd, Jan 19 2025

Andrew Howroyd, Table of n, a(n) for n = 2..500
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Main diagonal of A046653.

Cf. A000087 (distinguished face of any size).

B1nm := proc(n,m) # eq (4.15)
    local j ;
    if m>=2 and n>= m  then
        add((3*m-2*j-1)*(2*j-m)*(j-2)!*(3*n-j-m-1)!/(n-j)!/(j-m)!/(j-m+1)!/(2*m-j)!,j=m..min(n,2*m) ) ;
        %*m/(2*n-m)! ;
    else
        0 ;
    end if;
end proc:
B2wj := proc(w,j) # eq (8.21)
    local k ;
    if  w >= j and j>=1 and w >= 1 then
        add((2*k-j+1)*(k-1)!*(3*w-k-j)!/(k-j+1)!/(k-j)!/(2*j-k-1)!/(w-k)!,k=j..min(w,2*j-1) ) ;
        %*j/(2*w-j+1)! ;
    else
        0;
    end if;
end proc:
Brwj := proc(r,w,j) # eq. (8.21)
    local k ;
    if  w >= j and j>=1 and w>=1 and r > 1 then
        add((2*k-j)*(k-1)!*(3*w-k-j-1)!/((k-j)!)^2/(2*j-k)!/(w-k)!,k=j..min(w,2*j) ) ;
        %*j/(2*w-j)! ;
    else
        0 ;
    end if;
end proc:
Brnm := proc(r,n,m)
    if r = 1 then
        B1nm(n,m) ;
    elif r = 2 and type(n,'odd') and type (m,'even') then
        B2wj((n-1)/2,m/2) ;
    elif modp(n,r) <> 0 or modp(m,r) <> 0 then
        0;
    else
        Brwj(r,n/r,m/r) ;
    end if;
end proc:
L := proc(n,m) # eq. (6.7)
    add(numtheory[phi](s)*Brnm(s,n,m),s=numtheory[divisors](m)) ;
    %/m ;
end proc:
seq(L(n,2),n=2..40) ; # R. J. Mathar, Apr 13 2019

B1nm[n_, m_] := If[m >= 2 && n >= m, Sum[(3m - 2j - 1)(2j - m)(j - 2)! (3n - j - m - 1)!/(n - j)!/(j - m)!/(j - m + 1)!/(2m - j)!, {j, m, Min[n, 2m] }] m/(2n - m)!, 0];
B2wj[w_, j_]  := If[w >= j && j >= 1 && w >= 1, Sum[(2k - j + 1)(k - 1)! (3 w - k - j)!/(k - j + 1)!/(k - j)!/(2j - k - 1)!/(w - k)!, {k, j, Min[w, 2 j - 1] }] j/(2w - j + 1)!, 0];
Brwj[r_, w_, j_] := If[w >= j && j >= 1 && w >= 1 && r > 1 , Sum[(2k - j)(k - 1)! (3w - k - j - 1)!/((k - j)!)^2/(2j - k)!/(w - k)!, {k, j, Min[w, 2j]}] j/(2w - j)!, 0];
Brnm[r_, n_, m_] := Which[r == 1, B1nm[n, m], r == 2 && OddQ[n] && EvenQ[m], B2wj[(n - 1)/2, m/2], Mod[n, r] != 0 || Mod[m, r] != 0, 0, True, Brwj[r, n/r, m/r]];
L[n_, m_] := Sum[EulerPhi[s] Brnm[s, n, m], {s, Divisors[m]}]/m;
Table[L[n, 2], {n, 2, 30}] // Flatten (* Jean-François Alcover, Apr 05 2020, after R. J. Mathar *)

a(n) = n-=2; if(n<=0, n==0, (n*binomial(3*n\2, n\2) + binomial(3*n, 2*n+1))/(n*(n+1)) ) \\ Andrew Howroyd, Jan 19 2025

Reference gives generating functions.

a(n+2) = binomial(floor(3*n/2), floor(n/2))/(n+1) + binomial(3*n, 2*n+1)/(n*(n+1)) for n > 0. - Andrew Howroyd, Jan 19 2025

More terms from R. J. Mathar, Apr 13 2019

Name clarified by Andrew Howroyd, Jan 19 2025

cp's OEIS Frontend

A000087 Number of unrooted nonseparable planar maps with n edges and a distinguished face.

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A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 2*k edges which are invariant under a rotation of a 1/2 turn.

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A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with r*n edges and a fixed outer face of r*k edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

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Examples

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Crossrefs

Programs

Maple

PARI

Formula

Extensions

A046650 Number of sensed nonseparable (2-connected) planar maps with n edges and a distinguished face of size 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2n+1 edges and a fixed outer face of 2k edges which are invariant under a rotation of a 1/2 turn.

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with rn edges and a fixed outer face of rk edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).