[seqfan] Re: "half-Stirling numbers of the first kind"

[Vladimir Shevelev]

Dear SeqFans,

It appears that the problem 3) has a simple solution. Consider a permutation 
 (k_1,...,k_n) of numbers (1,...,n). A test when it is a permutation 
 of  the second indices in summands of hperm of square matrix A={a_(i,j)} of order n (let us call such a permutation a suitable one) is the following.
We distinguish two cases. 1) Both k_{n-1} and k_n differ from 1 and n. Then the permutation is suitable iff k_{n-1}<k_n;  
 2) Otherwise, the permutation is suitable iff  k_{n-1}=1 or (and) k_n=n.

Regards,
Vladimir



----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Tuesday, September 25, 2012 4:31
Subject: [seqfan] Re: "half-Stirling numbers of the first kind"
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Dear Olivier,
> 
> You are right, they are equivalent problems, and your 
> calculation is excellent. However, concretely in this problem, I 
> am not able to describe this constraint on the  relative 
> position of two first symbols, such that the problem 3) remains open.
> 
> Best regards,
> Vladimir
> 
> ----- Original Message -----
> From: Olivier Gerard <olivier.gerard at gmail.com>
> Date: Tuesday, September 25, 2012 2:22
> Subject: [seqfan] Re: "half-Stirling numbers of the first kind"
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> 
> > Dear Vladimir,
> > 
> > If I understand your idea correctly, this is equivalent to the 
> > followingproblem:
> > 
> > Counting permutations by cycles with a constraint on the 
> > relative position
> > of two first symbols (as
> > you are starting from a 2x2 matrix). Permutations with the 
> > pattern ... 1
> > ... 2 .... are easy to compute.
> > 
> > This gives the following triangle:
> > 
> > 0 | 0
> > 
> > 1 | 0 1
> > 
> > 3 | 1 1 1
> > 
> > 12 | 3 5 3 1
> > 
> > 60 | 12 24 17 6 1
> > 
> > 360 | 60 134 110 45 10 1
> > 
> > 2520 | 360 870 799 375 100 15 1
> > 
> > 20160 | 2520 6474 6489 3409 1050 196 21 1
> > 
> > 
> > It is not currently in the OEIS. If you agree, I propose to 
> > enter it in the
> > Encyclopedia with reference to this mail.
> > 
> > 
> > With my best regards,
> > 
> > 
> > Olivier GERARD
> > 
> > 
> > 
> > On Tue, Sep 25, 2012 at 2:42 PM, Vladimir Shevelev 
> > <shevelev at bgu.ac.il>wrote:
> > > Dear SeqFans,
> > >
> > > Let us define recursively half-permanent (hperm) of a square 
> > nxn (n>=2)
> > > matrix. For 2x2 matrix A with usual 2-index numeration 
> > elements, we put
> > >  hperm(A)=a_{11}*a_{22}. For 3x3 matrix, by the 
> expansion 
> > over its first
> > > row (without alternating signs),  and using the 
> > definition of
> > > half-permanent for 2x2 matrices, we define
> > >  
> > 
> hperm(A)=a_{11}*a_{22}*a_{33}+a_{12}*a_{21}*a_{33}+a_{13}*a_{21}*a_{32}.> In the same way, using the definition of half-permanent for 3x3 matrices,
> > > we define hperm(A) for 4x4 matrix, etc.
> > > Consider permutations of (1,2), (1,2,3), (1,2,3,4), etc. 
> given 
> > by the
> > > second indices in summands.  In case n=2, we have only 
> > permutation (1,2)
> > > with two cycles; in case
> > > n=3, we have 3 permutations (1,2,3),(2,1,3),(3,1,2) with 
> 3,2,1 
> > cycles> respectively; in case n=4 we have 12 permutations: one 
> > with 4 cycles, 3
> > > with 3 cycles, 5 with 2 cycles and 3 with one cycle. Thus we 
> > obtain the
> > > triangle (number of permutations of n elements over number 
> of 
> > cycles:> 1,2,3,...)
> > > (1)
> > > 0 1
> > > 1 1 1
> > > 3 5 3 1
> > > ....
> > > with row sums n!/2, n>=2.
> > > It is natural to call these numbers "half-Stirling of the 
> > first kind".
> > > Problems: 1) To coninue the triangle; 2) To find a GF for 
> half-
> > Stirling> numbers of the first kind; 3) Consider a permutation 
> > (k_1,...,k_n) of
> > > numbers (1,...,n). To find a test when it is a permutation 
> of 
> > the second
> > > indices in summands of hperm of square matrix A={a_(i,j)} of 
> > order n.
> > >
> > > Regards,
> > > Vladimir
> > >
> > >
> > >  Shevelev Vladimir‎
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> > 
> > _______________________________________________
> > 
> > Seqfan Mailing list - http://list.seqfan.eu/
> > 
> 
>  Shevelev Vladimir‎
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 

 Shevelev Vladimir‎