# [seqfan] Permutations along all diagonals and subdiagonals

Ron Hardin rhhardin at att.net
Tue Jul 27 04:05:18 CEST 2010

```Consider a N X M array with every diagonal or subdiagonal, if it has length L,
containing a permutation of 1..L

(thus the corners are all 1)

For instance a 5 X 4 array
1  1  1  1
2  3  3  2
2  4  4  2
2  3  3  2
1  1  1  1
(unique solution)

The number of solutions for NxM so far comes out, table with N=2.. and M=2..

N=2 1
N=3 2 1
N=4 1 2 1
N=5 2 5 2 1
N=6 1 10 1 2 1
N=7 2 13 26 65 2 1
N=8 1 26 1 626 1 2 1
N=9 2 61 290 9793 17690 4097 2 1
N=10 1 122 1 68122 1
N=11 2 221 2026 596161
N=12 1 442 1 3690242 1
N=13 2 925 16642 23378881
N=14 1 1850 1 178623226
N=15 2 3613 145162 1443811265
N=16 1 7226 1 11457775682
N=17 2 14621 1185922 91853884865
N=18 1 29242 1 704614184570
N=19 2 58141 9715690 5394132739009
N=20 1 116282 1 41233671609026
N=21 2 233245 80874050 315282132304321
N=22 1 466490 1 2433976660951802
N=23 2 931613 668377610 18833511341758401
N=24 1 1863226 1 145456660853820482
N=25 2 3729181 5516478530 1123458301034827201

where the counts are excess 1, ie there are no solutions for N=2 M=2

It seems there are no nXn solutions, at least for cases I can compute, up to
9x9.

I'm not sure what series, if any, to make of this.

rhhardin at mindspring.com
rhhardin at att.net (either)

```

More information about the SeqFan mailing list