a new problem related to partitions, dissections and polyominoes

[Max]

Wow!

Actually, that sequence is related to a quite popular research topic
in Russia called "threshold functions" but  somehow it's almost unkown
in English-speaking world. This sequence simply counts the number of
2-dimentional threshold functions.

I have a paper which I have been going to publish for already 5 years
and where I computed the exact formula for the number 2-dimentional
threshold functions on m X n grid. I can send an english version of my
paper to anybody interested. In turn I will be happy to hear
suggestions on what journal it would be better to submit and related
english references.

As of the sequence, it continues as

1, 7, 29, 87, 201, 419, 749, 1283, 2041, 3107, 4493, 6395, 8745,
11823, 15557, 20075, 25457, 32087, 39725, 48935, 59457, 71555, 85253,
101251, 119041, 139351, 161933, 187255, 215137, 246691, 280917,
319347, 361329, 407303, 457181, 511715, 570233, 634727, 704117,
779075, 859169, 946623, 1039373, 1140239, 1247617, 1362083, 1483685,
1614995, 1753473, 1901931, 2058501

Max

On 2/22/06, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Seqfans,  this arose from a question i received yesterday from a correspondent:
>
> %I A114043
> %S A114043 1,7,29,87,201
> %N A114043 Take an n X n square grid of points in the plane; a(n) = number of ways to divid
> e the points into two sets using a straight line.
> %C A114043 The line may not pass through any point. This is the "labeled" version - rotatio
> ns and reflections are not taken into account.
> %C A114043 8*binomial(2n,n) = 8*A000984(n) is a crude upper bound. Related to A001168.
> %O A114043 1,2
> %K A114043 nonn,nice,more
> %e A114043 Examples: the two sets are indicated by X's and o's.
> %e A114043 a(2) = 7:
> %e A114043 XX oX Xo XX XX oo oX
> %e A114043 XX XX XX Xo oX XX oX
> %e A114043 --------------------
> %e A114043 a(3) = 29:
> %e A114043 XXX oXX ooX ooo ooX ooo
> %e A114043 XXX XXX XXX XXX oXX oXX
> %e A114043 XXX XXX XXX XXX XXX XXX
> %e A114043 -1- -4- -8- -4- -4- -8- Total = 29
> %A A114043 Ugo Merlone (merlone(AT)econ.unito.it) and njas, Feb 22 2006
>
>
> I think it is quite interesting.  Maybe someone would like to
> compute some more terms!
>
> Although at first glance it seems to be quadratic in n,
> I believe - because of the connection with A001168 and
> other polyomino sequences that it is in fact exponential.
>
>