A295260 -id:A295260 - cp's OEIS frontend

A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 7, 5, 1, 1, 3, 5, 9, 12, 5, 1, 1, 4, 6, 18, 22, 30, 14, 1, 1, 4, 7, 21, 35, 52, 55, 14, 1, 1, 5, 8, 34, 51, 136, 140, 143, 42, 1, 1, 5, 9, 38, 70, 190, 285, 340, 273, 42, 1, 1, 6, 10, 55, 92, 368, 506, 1155, 969, 728, 132

Offset: 1

Andrew Howroyd, Feb 08 2024

The polygon prior to dissection will have n*(k-2)+2 sides.

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   2   2    3    3    4    4     5 ...
4  |  2   3   4    5    6    7    8     9 ...
5  |  2   7   9   18   21   34   38    55 ...
6  |  5  12  22   35   51   70   92   117 ...
7  |  5  30  52  136  190  368  468   775 ...
8  | 14  55 140  285  506  819 1240  1785 ...
9  | 14 143 340 1155 1950 4495 6545 12350 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns are A208355(n-1), A047749 (k=4), A369472 (k=5), A143546 (k=6), A143547 (k=8), A143554 (k=10), A192893 (k=12).

Cf. A070914 (rooted), A295224 (oriented), A295260 (unoriented), A369929, A370060 (achiral rooted at cell).

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {(if(n%2, u((n-1)/2, k, k\2), if(k%2, u(n/2-1, k, k-1), u(n/2, k, 1))))}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

T(n,k) = 2*A295260(n,k) - A295224(n,k).
T(n,2*k+1) = A370060(n,2*k+1).
T(n,2*k) = A369929(n,2*k-1).

A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 6, 7, 4, 1, 3, 11, 24, 24, 12, 1, 3, 17, 51, 89, 74, 27, 1, 4, 26, 109, 265, 371, 259, 82, 1, 4, 36, 194, 660, 1291, 1478, 891, 228, 1, 5, 50, 345, 1477, 3891, 6249, 6044, 3176, 733, 1, 5, 65, 550, 3000, 10061, 21524, 29133, 24302, 11326, 2282

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  3,   3;
1, 2,  6,   7,   4;
1, 3, 11,  24,  24,   12;
1, 3, 17,  51,  89,   74,   27;
1, 4, 26, 109, 265,  371,  259,  82;
1, 4, 36, 194, 660, 1291, 1478, 891, 228;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A001004.
Column k=3 is A003453.
Diagonals include A000207, A003449, A003450.
Cf. A295260, A295419, A295633.

\\ See A295419 for DissectionsModDihedral()
T=DissectionsModDihedral(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

A005419 Number of nonequivalent dissections of a polygon into n heptagons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 3, 16, 112, 1020, 10222, 109947, 1230840, 14218671, 168256840, 2031152928, 24931793768, 310420597116, 3912823963482, 49853370677834, 641218583442360, 8316918403772790, 108686334145327785, 1429927553582849256, 18927697628428129728, 251931892228273729375

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).

G. C. Greubel, Table of n, a(n) for n = 1..850
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Column k=7 of A295260.

p=7; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)

See Mathematica code.

a(n) ~ 2^(6*n - 1) * 3^(6*n + 1/2) / (sqrt(Pi) * n^(5/2) * 5^(5*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016

More terms from Robert A. Russell, Dec 11 2004

Name edited by Andrew Howroyd, Nov 20 2017

A290646 Number of dissections of an n-gon into 3- and 4-gons counted up to rotations and reflections.

Original entry on oeis.org

1, 2, 2, 7, 14, 53, 171, 691, 2738, 11720, 50486, 224012, 1005468, 4581815, 21093190, 98093226, 459986674, 2173599817, 10340539744, 49496519950, 238240366274, 1152543685463, 5601603835982, 27341242042238, 133977037982121, 658902522544060, 3251446102879398

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one triangle and one quadrangle.

Andrew Howroyd, Table of n, a(n) for n = 3..200
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Cf. A001004 (counted distinctly).

Cf. A001002, A290571, A295260, A295419.

(* See A295419 for DissectionsModDihedral. *)
DissectionsModDihedral[Boole[# == 3 || # == 4]& /@ Range[1, 30]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)

\\ See A295419 for DissectionsModDihedral.
DissectionsModDihedral(apply(v->v==3||v==4, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A290571 Number of dissections of an n-gon into 3- and 5-gons counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 7, 22, 60, 208, 695, 2566, 9451, 36158, 139574, 548347, 2174801, 8719651, 35244472, 143581782, 588858667, 2430036786, 10083626092, 42055927173, 176217259551, 741517642476, 3132564196880, 13281805256068, 56503895845238, 241135999611542

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

Cf. A001004, A290646, A295260.

\\ See A295419 for DissectionsModDihedral().
DissectionsModDihedral(apply(v->v==3||v==5, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

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A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

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A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

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A005419 Number of nonequivalent dissections of a polygon into n heptagons by nonintersecting diagonals up to rotation and reflection.

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A290646 Number of dissections of an n-gon into 3- and 4-gons counted up to rotations and reflections.

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A290571 Number of dissections of an n-gon into 3- and 5-gons counted up to rotations and reflections.

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