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Trace of sequence
Denote by
Denote by <math>N=\{0,1,2,...\}\,</math> the set of natural numbers and by <math>I_m=\{0,1,...,m-1\}\,</math> the set of natural numbers lesser than given natural number m. Lets <math>c=(c_0,c_1,...,c_{m-1})\,</math> a m-sequence of natural numbers and <math>p=max\{c_0,c_1,...,c_{m-1}\}\,</math> the greatest term of sequence c then the sequence
:<math>N=\{0,1,2,...\}\,</math>
the set of natural numbers and by
:<math>I_m=\{0,1,...,m-1\}\,</math>
the set of natural numbers lesser than given natural number m.
Lets
:<math>c=(c_0,c_1,...,c_{m-1})\,</math>
a m-sequence of natural numbers and
:<math>p=max\{c_0,c_1,...,c_{m-1}\}\,</math>
the greatest term of sequence c then the sequence
 
:<math>t(c)=(t_0,t_1,...,t_p)\,</math>
 
where <math>t_j,j\in I_{p+1}\,</math> denote number of terms of sequence c thats are equal at j, is called trace of c. Is clear that terms of trace fulfills the conditions
Is clear that terms of trace fulfills the conditions
 
:<math>t_0+t_1+...+t_p=m\,</math>
Claim:For each finite sequence <math>a\,</math> of natural numbers exists natural number n such that <math>t^n(a)\in H\,</math> in other words each sequence converges to H.
 
#Sequence <math>a\,</math> is of type <math>B\,</math> if its converge to H from B for example sequence (2,3) is of type B because
 
: <math>t^3((2,3))=(0,0,2)\in B\,</math>
 
#And sequences that converges to H from R are of type R for example the sequence (0) is of type R because
 
:<math>t^6((0))=(1,2)\in R\,</math>
 
My question is.Is my assumption true and if it is true how to decide of which type is any given finite sequence of natural numbers, can be done any programme or algorithmealgorithm. Thanks
 
My question is.Is my assumption true and if it is true how to decide of which type is any given finite sequence of natural numbers, can be done any programme or algorithme. Thanks
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ThisFirstly is not a complete solution, but it seems towe reduce the problem to an analysis of the cases <math>m\le2\,</math>.
I'll write <math>t^k\,</math> for the <math>k\,</math>-fold composition of <math>t\,</math> with itself, so that <math>t^1=t\,</math>, <math>t^2=t\circ t\,</math>, etc.
I'll also write <math>|{c}|\,</math> for the number of distinct elements of the <math>m\,</math>-tuple <math>c=(c_0,c_1,\dots,c_{m-1})\in\mathbb{N}^m\,</math>.