# [seqfan] about A001623 "Number of 3 X n normalized Latin rectangles."

Wouter Meeussen wouter.meeussen at telenet.be
Sun Oct 27 16:03:05 CET 2013

```the reference given as
“D. S. Stones, The many formulae for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.”
is also available as a pdf :
http://www.emis.ams.org/journals/EJC/Volume_17/PDF/v17i1a1.pdf

There, on pg 18, I see the formula for R_3_n
(translated into Mathematica)
Table[Sum[  n (n - 3)! (-1)^j 2^(n -i-j) i!/(n-i-j)! Binomial[3 i + j + 2, j], {i, 0, n}, {j, 0, n - i} ], {n, 3, 25}]
This easily produces 3 full lines (and confirms the existing 11 terms):

{1, 4, 46, 1064, 35792, 1673792, 103443808, 8154999232, 798030483328, 94866122760704, 13460459852344064,
2246551018310998016, 435626600453967929344, 97108406689489312301056, 24658059294992101453262848,
7075100096781964808223653888, 2277710095706779480096994066432, 817555425148510266964075644059648}

But I put it to you that the title is unclear:
what is counted are *Reduced* Latin rectangles, not normalised ones. See pg 2 in reference.

A Latin rectangle [L_n,k] is called normalised [R_n,k] if the first row is (0,1, . . . , n−1), and
reduced [R_n,k] if the first row is (0,1, . . . , n−1) and the first column is (0,1, . . . , k−1).

L_n,k = n! K_n,k = n! (n-1)! /(n-k)!    R_n,k

I will not correct\ammend this sequence myself since it is an old one,
and bears N.J.A. Sloane as autor.
Hence I consider this up for validation by a ‘senior’.

Wouter.
(know to goof occasionally  ;-))

```