[seqfan] Re: Stones On A Finite Triangular Lattice

[Leroy Quet]

--- On Fri, 9/3/10, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:

> 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.

Here -- I forgot to put the obvious -- you put each stone in a smaller triangle.


Here is some ascii art.

n = 4 case, two stones:

       /\
      /  \
      ----
     /\  /\
    /  \/  \
    --------
   /\  /\  /\
  /  \/ o\/  \
  ------------
 /\  /\  /\  /\
/ o\/  \/  \/  \
----------------

Leroy

> 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
> 
> 
> [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] )
> ]
> 
> 
>       
> 
> 
> 
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