[seqfan] Re: Several remarks on A088855

[jean-paul allouche]

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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>
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