# a too-short sequence related to Latin squares

hv at crypt.org hv at crypt.org
Fri Sep 3 19:16:52 CEST 2004

```"N. J. A. Sloane" <njas at research.att.com> wrote:
:Dear Seqfans,  is the following sequence already
:in the OEIS?  one more term would be helpful!
:Neil
:
:How many n X n squares are there, filled with
:the numbers 1...n, such that in row or column k,
:for all k = 1...n, the number k appears at least once?
:
:For n = 1 there is just 1,
:
:for n = 2 there seem to be 6
:namely
:
:11 11 12 12 12 21
:12 22 12 21 22 12

Naming the squares like:
AB
CD
.. this is:
A =1,D =2: 4 (2B 2C, ie 2 options for B times 2 options for C)
A =1,D!=2: 1 (1B 1C)
A!=1,D =2: 1 (1B 1C)
A!=1,D!=2: 0
==
6

For n = 3 and naming ABC/DEF/GHI, I make it:
A =1,E =2,I =3: 729 (3B 3C 3D 3F 3G 3H)
A =1,E =2,I!=3: 450 (3B 3D 2I 5(CF) 5(GH))
A =1,E!=2,I =3: 450
A!=1,E =2,I =3: 450
A =1,E!=2,I!=3: 196 (2E 2I 7(CDF) 7(BGH))
A!=1,E =2,I!=3: 196
A!=1,E!=2,I =3: 196
A!=1,E!=2,I!=3:  16 (2A 2E 2I 2(BCDFGH))
====
2683

No sequence in the database matches (1, 6, 2683); I don't think I'd want
to work out n = 4 without some programming.

Hugo

```