# a question

Max Alekseyev maxale at gmail.com
Wed Apr 4 21:47:20 CEST 2007

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Max

On 4/4/07, Augustine Munagi <aomunagi at gmail.com> wrote:
> Emeric,
> You might get a hint from the following communication.
>
> Discrete Mathematics, Volume 306, Issue 18, 28 September 2006, Pages 2234-2240.
>
> Augustine
>
>
> On 4/4/07, Emeric Deutsch <deutsch at duke.poly.edu> wrote:
> > Dear Seqfans,
> >
> > The following is known:
> > Given a partition p of n into k parts, [a_1^e_1,...,a_j^e_j]
> > (e_1 + e_2 + ... + e_j = k), the number b(p) of Dyck paths of
> > semilength n whose ascent lengths yield the partition p is
> >        b(p)=n!/[(n-k+1)!*e_1!*e_2!*...*e_j!].
> >
> > Apparently, we have
> >
> >      SUM(b(p)*SUM(e_i*binom(a_i + 1, 2), i=1..j) = binom(2n+1,n-1),
> >
> > where the outer sum is over all partitions p of n.
> >
> > For example, for n = 4, the partitions 4, 31, 22, 211, 1111 yield
> >      1*10 + 4*7 + 2*6 + 6*5 + 1*4 = 84.
> >
> > Has anybody seen this? Any idea for a proof? Any information?
> > Many thanks.
> >
> > Emeric
>

There are no cross-references between them, even though for the terms shown
the two sequences are the same:

A014692, n-th prime - (n-1):

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214, 219, 222, 223

A086969, Number of primes between p(n) and p(p(n)) inclusive:

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214

I don't quite understand how A086969 operates. If I plug in 1 for n I get:

p(1) = 2
p(p(1)) = p(2) = 3, and there are zero primes between 3 and 2. Am I missing

Andrew Plewe said:

There are no cross-references between them, even though for the terms shown
the two sequences are the same:

A014692, n-th prime - (n-1):

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214, 219, 222, 223

A086969, Number of primes between p(n) and p(p(n)) inclusive:

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214

I don't quite understand how A086969 operates. If I plug in 1 for n I get:

p(1) = 2
p(p(1)) = p(2) = 3, and there are zero primes between 3 and 2. Am I missing

Sorry, dumb mistake on my part. I derived the same thing as Ray Chandler
right about the time his message hit my inbox. Thanks!

-----Original Message-----

Andrew Plewe said:

There are no cross-references between them, even though for the terms shown
the two sequences are the same:

A014692, n-th prime - (n-1):

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214, 219, 222, 223

A086969, Number of primes between p(n) and p(p(n)) inclusive:

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52,
53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121,
126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188,
189, 198, 203, 208, 213, 214

I don't quite understand how A086969 operates. If I plug in 1 for n I get:

p(1) = 2
p(p(1)) = p(2) = 3, and there are zero primes between 3 and 2. Am I missing