# [seqfan] More results on Basket Weave tiling paths

David Scambler dscambler at bmm.com
Fri Feb 18 14:28:55 CET 2011

```The previous post on basket weave paths used a tiling with
alternating pairs of 2x1 and 1x2 tiles. Now this is
extended to triples of 3x1 and 1x3, or in general
alternating sets of m mx1 and 1xm tiles. Of course the 1x1
case is a normal square lattice.

I have counted unrestricted paths, paths weakly above x=y and
paths strictly above x=y, always using only North and East steps.

Also I have counted a few paths with different number of vertical versus
horizontal strands in the weave.

This hits quite a few existing sequences, especially Paul Barry's
sequences in Journal of Integer Sequences, Vol. 9 (2006),. 06.2.4.
"On Integer-Sequence-Based Constructions of. Generalized Pascal Triangles"
http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.pdf.

1) Tilings with m vertical and m horizontal weave strands

Unrestricted paths (0,0) to (mn, mn)
m
1	A000984	1,2,6,20,70,252,924,3432,12870,48620,184756,
2	A001850	1,3,13,63,321,1683,8989,48639,265729,1462563,8097453,
3	A069835	1,4,22,136,886,5944,40636,281488,
4	A084771	1,5,33,245,1921,15525,127905,1067925,
5	A084772	1,6,46,396,3606,33876,324556,3151896,
6	A098659	1,7,61,595,6145,65527,712909,7863667,
7	not OEIS	1,8,78,848,9766,116208,1411404,
8	not OEIS	1,9,97,1161,14721,192969,2582881,
9	not OEIS	1,10,118,1540,21286,304300,4443580,
10	not OEIS	1,11,141,1991,29761,460251,7272861,

Weakly above x=y  (see Paul Barry ref in these)
m
1	A000108	1,1,2,5,14,42,132,429,1430,4862,16796,
2	A001003	1,1,3,11,45,197,903,4279,20793,103049,518859,
3	A007564	1,1,4,19,100,562,3304,20071,124996,793774,5120632,
4	A059231	1,1,5,29,185,1257,8925,65445,491825,
5	A078009	1,1,6,41,306,2426,20076,171481,1500666,
6	A078018	1,1,7,55,469,4237,39907,387739,3858505,
7	A081178	1,1,8,71,680,6882,72528,788019,8766248,
8	A082147	1,1,9,89,945,10577,123129,1476841,
9	A082181	1,1,10,109,1270,15562,198100,2596645,
10	A082148	1,1,11,131,1661,22101,305151,4335711,

Strictly above x=y
m
1	A120588	1,1,1,2,5,14,42,132,429,1430,4862,
2	A155069	1,1,2,6,22,90,394,1806,8558,41586,206098,
3	not OEIS	1,1,3,12,57,300,1686,9912,60213,374988,2381322,
4	not OEIS	1,1,4,20,116,740,5028,35700,261780,

2) Tilings with v vertical and h horizontal weave strands

The widths of the tiles are adjusted so that the unit cell is still
a square, with sides m = max(v, h).

Unrestricted paths (0,0) to (mn, mn)
v x h
1x2	A085362	1,2,8,34,150,678,3116,14494,68032,321590,1528776,
2x1	A085362	1,2,8,34,150,678,3116,14494,68032,321590,1528776,

Weakly above x=y
v x h
1x2	A002212	1,1,3,10,36,137,543,2219,9285,39587,
2x1	A026375	1,3,11,45,195,873,3989,18483,86515,408105,1936881,

Strictly above x=y
v x h
1x2	A181768	1,1,2,5,15,51,188,731,
2x1	not OEIS	1,1,1,3,10,36,137,543,2219,

Any other VxH - not in OEIS.

(some matches differ in starting index from the OEIS sequence)

dave

2) Tilings with v vertical and h horizontal strands

Unrestricted paths (0,0) to (mn, mn)
v x h

1x2
A085362
1,2,8,34,150,678,3116,14494,68032,321590,1528776,
2x3
not OEIS
1,3,15,85,503,3049,18785,117115,736751,4667513,29737325,

Weakly above x=y

v x h

1x2
A002212
1,1,3,10,36,137,543,2219,9285,39587,
2x3
not OEIS
1,1,4,18,87,443,2347,12821,71739,409171,

Strictly above x=y

v x h

1x2
A181768
1,1,2,5,15,51,188,731,
2x3
not OEIS
1,1,3,11,46,212,1049,5463,

```