[seqfan] Re: Several remarks on A088855

[Vladimir Shevelev]

Hi Jean-Paul,

Thank you very much for
this information.
Unfortunately, most likely,
Avi Peretz for some reason
no longer works in mathematics.
My attempts to contact him
did not work.

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of jean-paul allouche [jean-paul.allouche at imj-prg.fr]
Sent: 23 November 2017 11:08
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Several remarks on A088855

Hi

If you look at
http://mathforum.org/kb/message.jspa?messageID=318721
there seems to be an old (?) email for Avi Peretz:
njk at netvision.net.il
[this email can be found in several sequences of the oeis]

There is also another place where he seems to have commented
a math. book
https://www.amazon.com/gp/profile/amzn1.account.AH3SJ4NXEC2XOHDSE67I2S5NQJ7Q/ref=cm_cr_arp_d_pdp?ie=UTF8

Also see
https://pubsonline.informs.org/doi/abs/10.1287/inte.1030.0061

May be you can contact him there (if this is the same person)

shalom
jean-paul



Le 22/11/17 à 10:29, Vladimir Shevelev a écrit :
> Dear Seq Fans,
>
> I suggest to name the sequence
> A010551 also "Peretz factorial" after
> Avi Peretz.
> In 2001, in  [formula in A010551]
> it seems he for the first time
> noted that A010551 is the number of
> permutations p of {1, 2, 3, ..., n}
> such that for every i, i and p(i) have
> the same parity, i.e., p(i) - i is even.
> Although he did not give a proof,
> it is easily reached by induction.
> Using Peretz factorial, I obtained
> the property 2) of entries of the
> triangle A088855.
>
> Unfortunately, I did not find any
> his paper. Maybe, anyone
> knows?
>
> Best regards,
> Vladimir
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
> Sent: 19 November 2017 21:17
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Several remarks on A088855
>
> Dear Seq Fans,
>
> A088855 is a fine triangle.
>
> I would like to show its remarkable
> properties which were not noted in A088855
> :
>
> 1) It is convenient for me to make the
> numbering of rows and columns as in
> Pascal's triangle,
> that is, starting with 0.
> So we change
> n by n+1, k by k+1 (although
> we retain the notation a(n,k) for the
> entries).
>
> Then the formula in A088855
> for entries a(n,k) of the triangle
> could be converted to an interesting
> form. Let us denote by n!^ the number
> A010551(n). Then
> a(n,k) = n!^/(k!^*(n-k)!^).
>
>   This expression we denote
> naturally by (binomial^)(n,k).
> In this form a(n,k) looks
> like as an analog of the number of
> k-combinations.
>   And this is justified
> according to the following
> reason
> :
>
> 2) A combinatorial sense
> :
> a(n,k) is the number of those
> k-combinations when we
> choose  k elements {n_1,..., n_k}
> from 1..n for which n_1 is odd,
> n_2 is even,..., n_k has the
> parity of k. For example,
> a(6,3) = 9, since we have
> only the following suitable
> triples
> :
> {1,2,3},{1,4,3},{1,6,3},
> {1,2,5},{1,4,5},{1,6,5},
> {3,2,5},{3,4,5},{3,6,5}.
>
> 3) Most of entries a(n,k)
> are obtained by Pascal rule
> :
> a(n,k) = a(n-1, k-1) + a(n-1,k),
> except for the cases when
> n is even and k is odd, but even
> in the exceptional cases we
> have an "almost Pascal rule"
> :
> a(n,k) =(n/(n+2))*
> (a(n-1, k-1) + a(n-1,k)), n>=2.
>
> 4) A generalization.
>
> Note that there are infinitely many
> triangles with close properties.
> If instead factorial to consider
> sequence A_t : multiply
> successively by 1,...,1,2,...,2,3,...,3,4, ...,4,...,
> n >= 1, a(0) = 1, where we have
> t times of every i, i>=1, then the case
> t=1 corresponds to Pascal's triangle,
> t=2 corresponds to A088855,
> t=3
> corresponds to the following triangle
>
> 1
> 1 . 1
> 1 . 1 . 1
> 1 . 1 . 1 . 1
> 1 . 2 . 2  . 2 . 1
> 1 . 2 . 4  . 4 . 2 . 1
> 1 . 2 . 4  . 8 . 4 . 2 . 1
> 1 . 3 . 6 . 12 .12 . 6 . 3 . 1
> ....
>
> such that in OEIS there are neither triangle
> nor its row sums, etc.
>
>
> Best regards,
> Vladimir
>
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> Seqfan Mailing list - http://list.seqfan.eu/
>
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