[seqfan] Re: b-files for apparently matching sequences?
Joerg Arndt
arndt at jjj.de
Wed Jun 30 10:34:44 CEST 2010
* Ron Hardin <rhhardin at att.net> [Jun 30. 2010 09:03]:
> Doing a survey of permutations constrained by ( i-A <= p(i) <= i+B),
> i=1..n, a lot of those sequences already exist, some not described
> with an obviously corresponding definition.
>
> Which raises the question what to do, if anything, with b-files for
> my permutation problem when the existing sequence doesn't have a
> b-file.
>
> The following are existing sequences (via a huge thicket of shell
> scripts) for various A,B.
Note these seqs are (by definition!) permanents of 0-1 matrices
with ones on diagonals close to main diagonal.
==> One script for all of them.
By symmetry we have P(A,B,n) == P(B,A,n)
where P(A,B,n) == number of length-n perms such
that (i-A <= p(i) <= i+B).
Persi Diaconis, Ronald Graham, Susan P.\ Holmes:
{Statistical problems involving permutations with restricted positions},
In: {State of the art in probability and statistics: Festschrift for Willem R. van Zwet},
Papers from the symposium held at the University of Leiden,
Leiden, March 23--26, 1999, pp.195-222, \bdate{2001}.
URL: \url{http://projecteuclid.org/euclid.lnms/1215090070}.
A closed from for the recurrence for P(A,B,n) would be a nice.
More general, a recurrence for the number of perms where
p(i)-i lies in a prescribed finite(!) set \in \ZZ.
(again, a script to enumerate these is easy to write,
I can do this if required).
>
> existing %N, corresponding %N in permutation sequence, offset
> equivalence, and (possibly still growing) b-file for the permutation
> sequence, with offsets adjusted for the existing sequence, stealing
> initial terms if necessary.
>
> What to do with the b-files?
>
> %N A020701 Pisot sequences E(3,5), P(3,5).
> %N xxxxxxx Number of permutations of 1..n with i-1<=p(i)<=i+1
> %C xxxxxxx (Empirical) a(n)=A020701(n-3)
> %H A020701 R. H. Hardin, <a href="b020701">Table of n,a(n) for n=0..97</a>
>
> %N A020695 Pisot sequence E(2,3).
> %N xxxxxxx Number of permutations of 1..n with i-1<=p(i)<=i+1
> %C xxxxxxx (Empirical) a(n)=A020695(n-2)
> %H A020695 R. H. Hardin, <a href="b020695">Table of n,a(n) for n=0..98</a>
>
> %N A002524 Number of permutations of length n within distance 2.
> %N xxxxxxx Number of permutations of 1..n with i-2<=p(i)<=i+2
> %C xxxxxxx (Empirical) a(n)=A002524(n)
> %H A002524 R. H. Hardin, <a href="b002524">Table of n,a(n) for n=0..100</a>
>
> %N A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-3<=p(i)<=i+2
> %C xxxxxxx (Empirical) a(n)=A072827(n)
> %H A072827 R. H. Hardin, <a href="b072827">Table of n,a(n) for n=1..100</a>
>
> %N A002526 Number of permutations of length n within distance 3.
> %N xxxxxxx Number of permutations of 1..n with i-3<=p(i)<=i+3
> %C xxxxxxx (Empirical) a(n)=A002526(n)
> %H A002526 R. H. Hardin, <a href="b002526">Table of n,a(n) for n=0..100</a>
>
> %N A072850 Number of permutations satisfying i-2<=p(i)<=i+4, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+2
> %C xxxxxxx (Empirical) a(n)=A072850(n)
> %H A072850 R. H. Hardin, <a href="b072850">Table of n,a(n) for n=1..100</a>
>
> %N A072854 Number of permutations satisfying i-3<=p(i)<=i+4, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+3
> %C xxxxxxx (Empirical) a(n)=A072854(n)
> %H A072854 R. H. Hardin, <a href="b072854">Table of n,a(n) for n=1..100</a>
>
> %N A072856 Number of permutations satisfying i-4<=p(i)<=i+4, i=1..n (permutations of length n within distance 4).
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+4
> %C xxxxxxx (Empirical) a(n)=A072856(n)
> %H A072856 R. H. Hardin, <a href="b072856">Table of n,a(n) for n=1..100</a>
>
> %N A072852 Number of permutations satisfying i-2<=p(i)<=i+5, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+2
> %C xxxxxxx (Empirical) a(n)=A072852(n)
> %H A072852 R. H. Hardin, <a href="b072852">Table of n,a(n) for n=1..100</a>
>
> %N A072855 Number of permutations satisfying i-3<=p(i)<=i+5, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+3
> %C xxxxxxx (Empirical) a(n)=A072855(n)
> %H A072855 R. H. Hardin, <a href="b072855">Table of n,a(n) for n=1..100</a>
>
> %N A154654 Number of permutations of length n within distance 5
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+5
> %C xxxxxxx (Empirical) a(n)=A154654(n)
> %H A154654 R. H. Hardin, <a href="b154654">Table of n,a(n) for n=1..100</a>
>
> %N A122189 Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1.
> %N xxxxxxx Number of permutations of 1..n with i-6<=p(i)<=i+1
> %C xxxxxxx (Empirical) a(n)=A122189(n+7)
> %H A122189 R. H. Hardin, <a href="b122189">Table of n,a(n) for n=1..107</a>
>
> %N A072853 Number of permutations satisfying i-2<=p(i)<=i+6, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-6<=p(i)<=i+2
> %C xxxxxxx (Empirical) a(n)=A072853(n)
> %H A072853 R. H. Hardin, <a href="b072853">Table of n,a(n) for n=1..100</a>
>
> %N A154655 Number of permutations of length n within distance 6
> %N xxxxxxx Number of permutations of 1..n with i-6<=p(i)<=i+6
> %C xxxxxxx (Empirical) a(n)=A154655(n)
> %H A154655 R. H. Hardin, <a href="b154655">Table of n,a(n) for n=1..100</a>
>
> %N A154656 Number of permutations of length n within distance 7
> %N xxxxxxx Number of permutations of 1..n with i-7<=p(i)<=i+7
> %C xxxxxxx (Empirical) a(n)=A154656(n)
> %H A154656 R. H. Hardin, <a href="b154656">Table of n,a(n) for n=1..100</a>
>
> %N A154657 Number of permutations of length n within distance 8
> %N xxxxxxx Number of permutations of 1..n with i-8<=p(i)<=i+8
> %C xxxxxxx (Empirical) a(n)=A154657(n)
> %H A154657 R. H. Hardin, <a href="b154657">Table of n,a(n) for n=1..68</a>
>
> %N A122265 The (1,10)-entry of the matrix M^n, where M is the 10 X 10 matrix {{0,1,0,0,0, 0,0,0,0,0},{0,0,1,0,0,0,0,0,0,0},{0,0,0,1,0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0}, {0,0,0,0,0,1,0,0,0,0},{0,0,0,0,0,0,1,0,0,0},{0,0,0,0,0,0,0,1,0,0},{0,0,0,0,0, 0,0,0,1,0},{0,0,0,0,0,0,0,0,0,1},{1,1,1,1,1,1,1,1,1,1}}.
> %N xxxxxxx Number of permutations of 1..n with i-9<=p(i)<=i+1
> %C xxxxxxx (Empirical) a(n)=A122265(n+9)
> %H A122265 R. H. Hardin, <a href="b122265">Table of n,a(n) for n=0..109</a>
>
> %N A154658 Number of permutations of length n within distance 9
> %N xxxxxxx Number of permutations of 1..n with i-9<=p(i)<=i+9
> %C xxxxxxx (Empirical) a(n)=A154658(n)
> %H A154658 R. H. Hardin, <a href="b154658">Table of n,a(n) for n=1..30</a>
>
>
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
>
>
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