increasing or decreasing subsequence

[bouyao]

Dear seqfan,

if we denote S(n) concatenation of numbers

How many increasing subsequence we can extract of S(n) ?

Examples S(n) = successive numbers
S(6) = 123456
increasing subsequence are :

1,2,3,4,5,6
1,2,3,4,56
1,2,3,456
1,2,34,56
1,2,3456
1,23,456
1,23456
12,34,56
12,3456
123,456
123456

for n=6 whe have 11 increasing subsequences (a(6)=11)

and our sequence is 1,2,3,5,7,11,15,22,51,...

if S(n) is concatenation of prime numbers
our sequence is 1,2,3,5,8,14,25

this sequence are not in OEIS
How many decreasing or increasing subsequence we can extract of S(n)
If S(n) is succesive numbers
If S(n) is prime numbers
If S(n) is odd or even numbers ?
we can apply several sequence 

Thanks

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