[seqfan] Stones On A Finite Triangular Lattice
Leroy Quet
q1qq2qqq3qqqq at yahoo.com
Fri Sep 3 20:40:19 CEST 2010
Take an equilateral triangle and subdivide it into n^2 smaller same-sized equilateral triangles -- where there are n horizontal rows of smaller triangles, and the kth row has 2k-1 triangles, the triangles alternating between pointing up and point down. So, we have a finite triangular lattice.
Let a(n) be the smallest number of stones (each stone being small enough to fit within one of the smaller triangles) such that every smaller triangle of the larger triangle is in at least one "row" shared by a stone.
A "row" may either run horizontally (0 degrees) or run at 120 degrees or run at 240 degrees. Each smaller triangle is contained in exactly 3 rows, one at each direction.
(By the way, it is easy to see that a(n+1) - a(n) = 0 or 1, for all n.)
Is {a(k)} in the OEIS? It must be.
Now, let b(n) = the number of ways that the a(n) non-distinguished stones can be arranged on the order-n triangular lattice, such that each smaller triangle is in at least one row shared by a stone.
And let c(n) = the number of ways that the a(n) non-distinguished stones can be arranged on the order-n triangular lattice, such that each smaller triangle is in at least one row shared by a stone, and such that each row has at most one stone.
(By "non-distinguished" stones, I mean that, say, if you have two stones in an arrangement, and you switch these two stones, then both of these arrangements count together only once.)
Are {b(k)} and/or {c(k)} in the OEIS?
Note, {b(k)} and {c(k)} are somewhat triangular-lattice analogs to the number of permutations of (1,2,3,...n), where the permutations are the way to place n stones on a square lattice with each square in at least one row or column shared by a stone.
Thanks,
Leroy Quet
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