[seqfan] Re: Stones On A Finite Triangular Lattice
Richard Guy
rkg at cpsc.ucalgary.ca
Fri Sep 3 21:54:49 CEST 2010
Problem 252 in The Inquisitive Problem Solver, MAA, 2002,
may be relevant. See pages 60, 87, 198, 276. R.
On Fri, 3 Sep 2010, Leroy Quet wrote:
> --- 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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