[seqfan] Re: Permutations with Windowed Affinity

Sat Aug 22 18:05:32 CEST 2015

Finally, nearness of 5 and window of 3 has the same same-recurrence behavior, albeit with a much larger order
/tmp/fdg
T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and every three consecutive elements having its maximum within 5 of its minimum

Table starts
...1.....1.....1.....1.....1....1....1....1...1...1..1.1.1.1
...2.....2.....2.....2.....2....2....2....2...2...2..2.2.2..
...3.....6.....6.....6.....6....6....6....6...6...6..6.6....
...5....14....24....24....24...24...24...24..24..24.24......
...8....31....78...120...120..120..120..120.120.120.........
..13....73...230...504...720..720..720..720.720.............
..21...160...506...930..1560.2400.2400.2400.................
..34...357..1128..1794..2352.3552.5424......................
..55...814..2641..3852..4704.5484...........................
..89..1836..6655..9246.10946................................
.144..4140.17261.24613......................................
.233..9379.46066............................................
.377.21163..................................................
.610........................................................

Empirical for column k:
k=1: a(n)=a(n-1)+a(n-2)
k=2: [order 13]
k=3: [order 62]
k=4: [same order 62] for n>74
k=5: [same order 62] for n>74
k=6: [same order 62] for n>76
k=7: [same order 62] for n>78

Some.solutions.for.n=10.k=4..
..0....4....1....0....1....0....0....0....0....4....1....3....1....2....0....0..
..1....1....3....2....0....3....2....5....1....0....3....5....3....0....5....1..
..2....0....2....1....2....2....1....1....3....2....2....4....0....1....1....3..
..3....3....0....3....4....1....3....6....2....1....0....0....2....4....2....2..
..5....2....4....4....3....4....4....2....4....3....4....1....4....5....3....7..
..8....5....5....7....5....6....5....3....7....5....5....2....7....3....7....5..
..4....6....6....9....6....7....8....4....9....6....8....6....8....6....4....4..
..9....7....9....6....7....8....7....7....6....9....7....7....6....7....8....6..
..6....8....7....5....9....5....9....8....5....7....6....8....5....9....6....9..
..7....9....8....8....8....9....6....9....8....8....9....9....9....8....9....8..

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

      From: Ron Hardin <rhhardin at att.net>
 To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu> 
 Sent: Friday, August 21, 2015 12:43 PM
 Subject: [seqfan] Re: Permutations with Windowed Affinity

Nearness of 4 and window of 4 has the same recurrence behavior. 

/tmp/fdh
T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and every four consecutive elements having its maximum within 4 of its minimum

Table starts
...1...1...1...1...1...1...1...1...1...1..1.1.1.1
...2...2...2...2...2...2...2...2...2...2..2.2.2..
...3...6...6...6...6...6...6...6...6...6..6.6....
...5..14..24..24..24..24..24..24..24..24.24......
...8..31..78.120.120.120.120.120.120.120.........
..11..34..60..72.144.144.144.144.144.............
..17..39..50..54..60.108.108.108.................
..25..46..52..54..54..60.108.....................
..37..64..70..72..72..72.........................
..57.104.116.120.120.............................
..84.161.184.192.................................
.127.249.292.....................................
.191.385.........................................
.284.............................................

Empirical for column k:
k=1: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7)
k=2: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18
k=3: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18
k=4: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18
k=5: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18
k=6: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18
k=7: a(n)=a(n-1)+a(n-3)-a(n-4)+2*a(n-5)-a(n-6)+a(n-7) for n>18

Some.solutions.for.n=7.k=4..
..0....0....1....0....0....0....0....3....1....0....0....0....0....1....1....0..
..1....1....0....1....4....1....1....0....0....1....2....1....1....0....0....1..
..3....3....3....2....1....3....2....1....3....3....1....2....3....4....4....4..
..4....2....4....3....3....4....4....4....2....2....4....3....2....2....3....2..
..2....5....2....5....5....5....5....2....4....4....3....4....4....3....2....3..
..5....4....5....6....2....6....3....5....5....5....5....5....6....5....5....5..
..6....6....6....4....6....2....6....6....6....6....6....6....5....6....6....6..

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

      From: Ron Hardin <rhhardin at att.net>
 To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu> 
 Sent: Thursday, August 20, 2015 6:44 PM
 Subject: [seqfan] Re: Permutations with Windowed Affinity

Substituting a nearness of 4 for a nearness of 3 gives a similar everybody-has-the-same-recurrence but of higher order and larger starting point
/tmp/fdf
T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and every three consecutive elements having its maximum within 4 of its minimum

Table starts
...1....1....1....1....1...1...1...1...1...1..1.1.1.1
...2....2....2....2....2...2...2...2...2...2..2.2.2..
...3....6....6....6....6...6...6...6...6...6..6.6....
...5...14...24...24...24..24..24..24..24..24.24......
...8...31...78..120..120.120.120.120.120.120.........
..13...56..110..168..288.288.288.288.288.............
..21..104..169..204..276.456.456.456.................
..34..208..301..348..374.488.768.....................
..55..418..616..696..732.754.........................
..89..873.1373.1601.1673.............................
.144.1772.2908.3476..................................
.233.3545.5908.......................................
.377.7103............................................
.610.................................................

Column 1 is A130137(n-1)

Empirical for column k:
k=1: a(n)=a(n-1)+a(n-2)
k=2: a(n)=a(n-1)+a(n-2)+7*a(n-5)+2*a(n-6)+a(n-7)+5*a(n-8)-4*a(n-9)-3*a(n-10)-3*a(n-11)-4*a(n-12)-4*a(n-14)-a(n-15)+3*a(n-16)-a(n-17)+a(n-19)
k=3: [same order 19] for n>27
k=4: [same order 19] for n>27
k=5: [same order 19] for n>29
k=6: [same order 19] for n>31
k=7: [same order 19] for n>33

Some.solutions.for.n=7.k=4..
..1....1....4....0....0....3....1....0....4....0....0....0....2....1....1....0..
..3....0....2....3....1....1....0....1....0....1....2....3....0....0....4....1..
..0....2....0....1....4....0....2....2....1....4....1....1....4....4....0....3..
..4....4....1....5....5....4....4....3....2....3....5....2....1....2....2....2..
..2....6....3....2....3....2....3....6....5....6....4....5....3....5....3....6..
..5....5....5....6....2....6....6....4....6....2....3....6....5....6....5....4..
..6....3....6....4....6....5....5....5....3....5....6....4....6....3....6....5..
 rhhardin at mindspring.com rhhardin at att.net (either)

      From: Ron Hardin <rhhardin at att.net>
 To: "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu> 
 Sent: Wednesday, August 19, 2015 8:07 PM
 Subject: [seqfan] Permutations with Windowed Affinity

Permutations of 0..n-1 that put elements that start close together, close together at the end.

Shift initial 0..n-1 by as much as +-k in the permutation.
In each consecutive 3 elements, the maximum-minimum is 3 or less.
A couple of things are surprising -
1.  Every column of T(n,k) has the same recurrence albeit with a different lower limit
2.  plateaus in the rows of T(n,k)
I have yet to try what windows and limits other than 3 produce.

/tmp/fde
T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and every three consecutive elements having its maximum within 3 of its minimum

Table starts
....1....1....1....1...1...1...1...1...1...1...1..1..1..1..1..1.1.1.1
....2....2....2....2...2...2...2...2...2...2...2..2..2..2..2..2.2.2..
....3....6....6....6...6...6...6...6...6...6...6..6..6..6..6..6.6....
....5...14...24...24..24..24..24..24..24..24..24.24.24.24.24.24......
....7...14...18...36..36..36..36..36..36..36..36.36.36.36.36.........
...11...16...18...20..36..36..36..36..36..36..36.36.36.36............
...16...22...24...24..27..48..48..48..48..48..48.48.48...............
...25...36...40...40..40..49..80..80..80..80..80.80..................
...37...56...64...64..64..64..76.128.128.128.128.....................
...57...85..100..100.100.100.100.120.200.200.........................
...85..125..144..144.144.144.144.144.168.............................
..130..189..216..216.216.216.216.216.................................
..195..285..324..324.324.324.324.....................................
..297..434..496..496.496.496.........................................
..447..655..748..748.748.............................................
..679..993.1136.1136.................................................
.1024.1499.1712......................................................
.1553.2271...........................................................
.2345................................................................

Column 1 is A130137(n-1)

Empirical for column k:
k=1: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4)
k=2: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>12
k=3: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>12
k=4: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>12
k=5: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>12
k=6: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>12
k=7: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>13

Some.solutions.for.n=7.k=4..
..1....0....0....3....1....0....0....0....0....0....0....0....1....0....1....2..
..0....2....1....0....0....1....1....1....1....2....1....2....0....1....0....0..
..2....1....2....1....3....2....2....2....2....1....2....1....2....2....3....1..
..3....3....3....2....2....4....3....4....3....3....4....4....3....4....2....3..
..4....4....4....4....4....3....5....5....4....4....3....3....5....5....5....4..
..6....5....6....5....5....6....4....6....5....6....5....6....6....3....4....6..
..5....6....5....6....6....5....6....3....6....5....6....5....4....6....6....5..
 rhhardin at mindspring.com rhhardin at att.net (either)

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