The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Marko Riedel (See also Marko Riedel's wiki page)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A376808

Number of non-isomorphic colorings of a toroidal n X n grid using any number of swappable colors.
(history; published version)

#16 by Marko Riedel at Sat Oct 05 13:41:16 EDT 2024

STATUS

editing

proposed

#15 by Marko Riedel at Sat Oct 05 13:40:37 EDT 2024

LINKS

Marko Riedel, <a href="/A376808/a376808.maple.txt">Maple code for sequence by PGE.</a>

STATUS

proposed

editing

#14 by Marko Riedel at Sat Oct 05 13:32:33 EDT 2024

STATUS

editing

proposed

#13 by Marko Riedel at Sat Oct 05 13:32:09 EDT 2024

FORMULA

T(n,k) = Sum_{Q=1..n*n^2} (1/(n^2*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.

STATUS

proposed

editing

#3 by Marko Riedel at Fri Oct 04 18:46:16 EDT 2024

STATUS

editing

proposed

#2 by Marko Riedel at Fri Oct 04 18:44:43 EDT 2024

NAME

allocated for Marko Riedel

Number of non-isomorphic colorings of a toroidal n X n grid using any number of swappable colors.

DATA

1, 9, 2387, 655089857, 185543613289205809, 106103186941524316132396201360, 218900758256599151027392153440612298654753249, 2689595989958732045849530682270318547733917269644639109073775285

OFFSET

1,2

COMMENTS

Two colorings are equivalent if there is a permutation of the colors that takes one to the other in addition to translational symmetries on the torus. (Power Group Enumeration.) Maximum number of colors is n * n.

REFERENCES

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

LINKS

Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>

FORMULA

T(n,k) = Sum_{Q=1..n*n} (1/(n^2*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.~

EXAMPLE

For the 2x2 we find

+-+-+ +-+-+ +-+-+ +-+-+ +-+-+

|X|X| |X|X| |X|X| |X| | |X| |

+-+-+ +-+-+ +-+-+ +-+-+ +-+-+

|X|X| |X| | | | | |X| | | |X|

+-+-+ +-+-+ +-+-+ +-+-+ +-+-+

+-+-+ +-+-+ +-+-+ +-+-+

|X|Y| |X| | |X| | |X|Y|

+-+-+ +-+-+ +-+-+ +-+-+

| | | |Y| | | |Y| |Z| |

+-+-+ +-+-+ +-+-+ +-+-+

CROSSREFS

Cf. A294791, A294792, A294793, A294794. a(n) = A295197(n,n)

KEYWORD

allocated

nonn

AUTHOR

Marko Riedel, Oct 04 2024

STATUS

approved

editing

#1 by Marko Riedel at Fri Oct 04 18:44:43 EDT 2024

NAME

allocated for Marko Riedel

KEYWORD

allocated

STATUS

approved

A376749

Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.
(history; published version)

#15 by Marko Riedel at Fri Oct 04 17:37:23 EDT 2024

STATUS

editing

proposed

A376748

Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.
(history; published version)

#14 by Marko Riedel at Fri Oct 04 17:37:11 EDT 2024

STATUS

editing

proposed

A376747

Number of non-isomorphic colorings of a toroidal n X n grid using exactly two swappable colors.
(history; published version)

#11 by Marko Riedel at Fri Oct 04 17:37:00 EDT 2024

STATUS

editing

proposed

Discussion

Fri Oct 04

17:51

Marko Riedel: BTW the OEIS knows when a sequence is a triangular array so maybe there should be an advanced search option to include diagonals in the search. That way I would have found those sequences yesterday.