Given positive interger N, now many non-decreasing sequences of length

  • Thread starter Dragonfall
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Given positive interger N, now many non-decreasing sequences of length N are there whose entries are less than N?
 
Re: Sequence

are the entries positive integers?
 
1,028
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Yes .
 
424
3
Re: Sequence

This isn't in the homework section so I'm assuming it isn't homework.

Let A be the set of sequences of the type in the post.
Let B be the set of sequences of N+1 non-negative integers with sum N-2.
A and B are in bijection by associating a sequence [itex]a_1,a_2, \ldots ,a_N[/itex] of A to [itex]b_1,b_2,\ldots,b_{N+1}[/itex] defined by [itex]b_i = a_i - a_{i-1}[/itex] where we let [itex]a_0 = 1, a_{N+1}=N[/itex]. The inverse is given by associating a sequence [itex]b_1,b_2,\ldots,b_{N+1}[/itex] of B to [itex]a_1,\ldots,a_N[/itex] defined by [itex]a_{i+1} = a_{i}+b_{i+1}[/itex] where we let [itex]a_0 = 1[/itex].

Thus we can just count B which has
[tex]\binom{(N+1)+(N-2)-1}{(N+1)-1} = \binom{2N-2}{N}[/tex]
elements.
 
1,028
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Re: Sequence

Wow I would never have thought of that. Thanks. Unfortunately it grows too fast for my purposes.
 

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