A370060 -id:A370060 - cp's OEIS frontend

A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 6, 1, 1, 1, 1, 3, 5, 7, 10, 1, 1, 1, 1, 4, 5, 16, 12, 20, 1, 1, 1, 1, 4, 7, 18, 31, 30, 35, 1, 1, 1, 1, 5, 7, 31, 35, 102, 55, 70, 1, 1, 1, 1, 5, 9, 34, 64, 136, 213, 143, 126, 1

Offset: 0

Andrew Howroyd, Feb 07 2024

T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.

			Array begins:
===============================================
n\k| 1  2   3   4    5    6    7    8     9 ...
---+-------------------------------------------
0  | 1  1   1   1    1    1    1    1     1 ...
1  | 1  1   1   1    1    1    1    1     1 ...
2  | 1  1   1   1    1    1    1    1     1 ...
3  | 1  2   2   3    3    4    4    5     5 ...
4  | 1  3   3   5    5    7    7    9     9 ...
5  | 1  6   7  16   18   31   34   51    55 ...
6  | 1 10  12  31   35   64   70  109   117 ...
7  | 1 20  30 102  136  296  368  651   775 ...
8  | 1 35  55 213  285  663  819 1513  1785 ...
9  | 1 70 143 712 1155 3142 4495 9304 12350 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
Wikipedia, Fuss-Catalan number.
Wikipedia, Noncrossing partition.

Columns are: A000012, A001405(n-1), A047749 (k=3), A369930 (k=4), A143546 (k=5), A143547 (k=7), A143554 (k=9), A192893 (k=11).

Cf. A070914, A303694, A303929, A361236, A361239, A370060, A370062.

\\ u(n,k,r) are Fuss-Catalan numbers.
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
T(n, k)={if(n==0, 1, if(k%2, if(n%2, 2*u(n\2, k, (k+1)/2), u(n/2, k, 1) + u(n/2-1, k, k)), e(n, k) + if(n%2, u(n\2, k, k/2)))/2)}

T(n,k) = 2*A303929(n,k) - A303694(n,k).

T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).

A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 52, 140, 340, 969, 2394, 7084, 17710, 53820, 135720, 420732, 1068012, 3362260, 8579560, 27343888, 70068713, 225568798, 580034052, 1882933364, 4855986044, 15875338990, 41043559340, 134993766600

Offset: 1

Robert A. Russell, Jan 23 2024

A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Column k=5 of A370060.

Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A002293 (rooted), A047749 {4,oo}, A143546 {6,oo}.

p=5; Table[If[EvenQ[n],Binomial[(p-1)n/2,n/2]/((p-2)n/2+1),If[OddQ[p],(p-1)Binomial[(p-1)n/2-1,(n-1)/2]/((p-2)n+1),p Binomial[(p-1)n/2-1/2,(n-1)/2]/((p-2)n+2)]],{n,35}]

For n even, a(n) = C(2n,n/2)/(3n/2+1).
For n odd, a(n) = 4*C(2n-1,(n-1)/2)/(3n+1).
a(n+2)/a(n) ~ 256/27. a(2m+1)/a(2m) ~ 32/9; a(2m)/a(2m-1) ~ 8/3.
a(n) = 2*A005040(n) - A005038(n) = A005038(n) - 2*A369471(n) = A005040(n) - A369471(n).
G.f.: G(z^2)+z*G(z^2)^2, where G(z)=1+z*G(z)^4, the generating function for A002293.
a(2m) = A002293(m) ~ (4^4/3^3)^m*sqrt(4/(2*Pi*(3*m)^3)). - Robert A. Russell, Jul 15 2024

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.

Original entry on oeis.org

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).

A370061 Number of achiral dissections of a polygon into n hexagons by nonintersecting diagonals rooted at a cell.

Original entry on oeis.org

1, 1, 4, 6, 26, 45, 204, 380, 1771, 3450, 16380, 32886, 158224, 324632, 1577532, 3290040, 16112057, 34034715, 167710664, 357919100, 1772645420, 3815041230, 18974357220, 41124015036, 205263418941, 447534498320, 2240623268512, 4910258796240, 24648785802336, 54257308779600

Offset: 1

Andrew Howroyd, Feb 08 2024

nonn

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

Column k=6 of A370060.

a(n)=my(m=n\2); if(n%2==0, 6*binomial(5*m+1, m-1)/(5*m+1), 4*binomial(5*m+4, m)/(5*m+4))

a(2*n) = 6*binomial(5*n+1, n-1)/(5*n+1); a(2*n+1) = 4*binomial(5*n+4, n)/(5*n+4).

cp's OEIS Frontend

A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

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A369472 Number of achiral polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}.

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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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A370061 Number of achiral dissections of a polygon into n hexagons by nonintersecting diagonals rooted at a cell.

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