Sasha Kolpakov - cp's OEIS frontend

User: Sasha Kolpakov

Sasha Kolpakov has authored 4 sequences.

A292207 Number of unrooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges).

2, 3, 16, 133, 1440, 22076, 401200, 8523946, 206375088, 5611089408, 169259764912, 5610386295418, 202710195084400, 7929759557219228, 333909047017798272, 15059194651009154172, 724232293050284717248

Offset: 1

Sasha Kolpakov, Sep 11 2017

nonn

Equivalently, the number of unrooted bicolored triangulations with 2*n triangles (and thus 3*n edges).
Equivalently, the number of pairs of permutations (alpha,sigma) up to simultaneous conjugacy on a set of size 3*n with alpha^3=sigma^3=1, acting transitively and without fixed points.
There is no recurrence relation known for this sequence.

Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.

Unrooted version of A292187.

Offset edited by Andrey Zabolotskiy, Jan 17 2025

A292206 Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).

Original entry on oeis.org

1, 2, 7, 36, 365, 5250, 103801, 2492164, 70304018, 2265110191, 82013270998, 3295691020635, 145553281837454, 7008046130978980, 365354356543414133, 20504381826687810441, 1232562762503125498772, 79012106044626365750974, 5380476164948914549410335, 387882486153123498708054879

Offset: 0

Sasha Kolpakov, Sep 11 2017

nonn

Equivalently, the number of unrooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges).

Equivalently, the number of pairs (alpha,sigma) of permutations on a set of size 4*n up to simultaneous conjugacy such that alpha (resp. sigma) has only cycles of length 2 (resp. 4) and the subgroup generated by them acts transitively.

			For n = 1, a(n) = 2:
1) the figure-eight map on a sphere (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the quadrangulation of a sphere created by a 2-edge path (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((12)(34), (1234));
2) the map on a torus consisting of two non-homotopic nontrivial loops (1 vertex, which has degree 4, and 2 edges) <-> its dual map, which is the same map again (it bounds 1 region, which has 4 boundary segments, even though they are formed by only 2 different edges) <-> the conjugacy class of the pair of permutations ((13)(24), (1234)).

Andrew Howroyd, Table of n, a(n) for n = 0..300
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.

Column 4 of A380626.
Unrooted version of A292186.
Cf. A268556.

Inverse Euler transform of A268556. - Andrew Howroyd, Jan 29 2025

Edited by Andrey Zabolotskiy, Jan 17 2025

a(0)=1 prepended and a(18) onwards from Andrew Howroyd, Jan 29 2025

A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges), with a(0) = 1.

Original entry on oeis.org

1, 2, 12, 112, 1392, 21472, 394752, 8421632, 204525312, 5572091392, 168331164672, 5585571889152, 201973854584832, 7905697598963712, 333049899230625792, 15025907115679875072, 722841343143300759552, 36935846945562562527232, 1997902532753538016346112, 114050521905958855289864192, 6852141240070150728132329472

Offset: 0

Sasha Kolpakov, Sep 11 2017

Equivalently, the number of rooted bicolored triangulations with 2*n triangles (and thus 3*n edges) for n > 0.
Equivalently, the number of pairs of permutations (alpha,sigma) up to simultaneous conjugacy on a pointed set of size 3*n with alpha^3=sigma^3=1, acting transitively and without fixed points, for n > 0.
This is also the S(3, -5, 1) sequence of Martin and Kearney, if the offset is set to 1.
This sequence is not D-finite (or holonomic).

Sasha Kolpakov, Table of n, a(n) for n = 0..119
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318; arXiv:1103.4936 [math.CO], 2011.

Cf. A000698, A062980, A292186.

from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n == 0 else (3*n - 1)*a(n - 1) + sum([a(k)*a(n - k - 1) for k in range(1, n)])
[a(n) for n in range(21)]

a(0)=1, a(1)=2, a(n) = 3*n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1) for n>=2.
From Peter Bala, Sep 01 2023: (Start)
The o.g.f. A(x) = 1 + 2*x + 12*x^2 + 112*x^3 + 1392*x^4 + 21472*x^5 + 394752*x^6 + ... satisfies the Riccati differential equation (3*x^2)*A'(x) = -1 + (1 - x)*A(x) - x*A(x)^2 with A(0) = 1.
O.g.f. as a continued fraction of Stieltjes type: A(x) = 1/(1 - 2*x/(1 - 4*x/(1 - 5*x/(1 - 7*x/(1 - 8*x/(1 - 10*x/(1 - ... ))))))).
Also A(x) = 1/(1 + 2*x - 4*x/(1 - 2*x/(1 - 7*x/(1 - 5*x/(1 - 10*x/(1 - 8*x/(1 - ... ))))))). (End)

Edited by Andrey Zabolotskiy, Jan 23 2025

A292186 Number of rooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges), with a(0) = 1.

Original entry on oeis.org

1, 3, 24, 297, 4896, 100278, 2450304, 69533397, 2247492096, 81528066378, 3280382613504, 145009234904922, 6986546222800896, 364418301804218028, 20459842995693256704, 1230262900677124568397, 78884016707711348637696, 5372823210133041283250178, 387394283866652086938107904

Offset: 0

Sasha Kolpakov, Sep 11 2017

Equivalently, the number of rooted quadrangulations of oriented surfaces with n quadrangles (and thus 2*n edges) for n > 0.
Equivalently, the number of pairs (alpha,sigma) of permutations on a pointed set of size 4*n up to simultaneous conjugacy such that alpha (resp. sigma) has only cycles of length 2 (resp. 4) and the subgroup generated by them acts transitively, for n > 0.
This is also the S(4, -6, 1) sequence of Martin and Kearney, if the offset is set to 1.
This sequence is not D-finite (or holonomic).

Sasha Kolpakov, Table of n, a(n) for n = 0..119
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, Discrete Mathematics, 342 (2019), 1415-1433; arXiv:1708.03842 [math.CO], 2017-2019.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318; arXiv:1103.4936 [math.CO], 2011.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Column 4 of A380622.

Cf. A172455, A292187, A128709.

from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n == 0 else (4*n - 2)*a(n - 1) + sum([a(k)*a(n - k - 1) for k in range(n)])
[a(n) for n in range(21)]

a(0)=1, a(1)=3, a(n) = 4*n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1) for n>=2.
From Peter Bala, Aug 22 2023: (Start)
The o.g.f. A(x) = 1 + 3*x + 24*x^2 + 297*x^3 + 4896*x^4 + 100278*x^5 + 2450304*x^6 + ... satisfies the Riccati differential equation (4*x^2)*A'(x) = -1 + (1 - 2*x)*A(x) - x*A(x)^2 with A(0) = 1.
O.g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x/(1 - 5*x/(1 - 7*x/(1 - 9*x/(1 - ... - (2*n+1)*x/(1 - ... )))))).
Also A(x) = 1/(1 + 2*x - 5*x/(1 - 3*x/(1 - 9*x/(1 - 7*x/(1 - ... - (4*n+1)*x/(1 - (4*n-1)*x/(1 - ... ))))))). (End)

Edited by Andrey Zabolotskiy, Jan 23 2025

cp's OEIS Frontend

User: Sasha Kolpakov

A292207 Number of unrooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2*n vertices (and thus 3*n edges).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A292206 Number of unrooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2*n vertices (and thus 3*n edges), with a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

Extensions

A292186 Number of rooted unlabeled connected four-regular maps on a compact closed oriented surface with n vertices (and thus 2*n edges), with a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

Extensions

A292207 Number of unrooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges).

A292187 Number of rooted unlabeled bipartite cubic maps on a compact closed oriented surface with 2n vertices (and thus 3n edges), with a(0) = 1.