OEIS programs

[hv at crypt.org]

hv at crypt.org wrote:

>Antti Karttunen <antti.karttunen at gmail.com> wrote:
>[...]
>:If one considers the same process in factorial base http://www.research.att.com/~njas/sequences/A007623
>:then (a similar) sequence can be made infinite. But then only few of these terms
>:can be used, because there is an additional condition that the nth digit from
>:the right can be at most n. So only 1, 221, 233321, 323321, 332321, 333221, etc. remain.
>:and the number of digits must be a triangular number.
>
>The last clause is incorrect: 4444221 is possible.
>
>  
>
Yes, I realized this in my morning shower (sometime after 14:00 .-),
but wanted to spare SeqFans from the flurry of corrections...
The reason for the mistake was that lately I have geared my attention to 
the subset :
http://www.research.att.com/~njas/sequences/A071158
( http://www.research.att.com/~njas/sequences/A071157 gives a better 
description)
in where my statement would be true.
(because 4 cannot occur unless there are also 3's present).

Now I wonder whether, if such an additional restriction is required,
i.e. the terms are subset of A071158, these map to any interesting
set of Dyck paths or other Catalan structures?

Actually, I'm just finishing with this sequence (I will send this to 
Neil in a SEPARATE batch,
not to be extracted from this mail!), where, although on the surface, it 
seems just
a digit manipulation in factorial base, it really is about the fixed 
points of
certain automorphism of Dyck paths:

%I A126301
%S A126301 0,1,23211,2432211,2354543212221,335465432122211
%N A126301 A071158-codes for the fixed points of Vaille's 1997 bijection 
on Dyck paths.
%D A126301 J. Vaill\'{e}, Une Bijection Explicative de Plusieurs 
Proprietes Remarquables des Ponts, European J. Combin. 18 (1997), no. 1, 
117-124.
%C A126301 Vaille gives the terms a(2)-a(4) on the last page of the 1997 
paper. Note that this sequence migh be finite, for two reasons: (a) 
there are no more fixed points after some limit (the next one after a(5) 
must have at least 19 digits. All the terms must be of odd length). (b) 
some of the fixed points would require digits higher than "9", in which 
case they cannot presented unambiguously in decimal. However, the 
sequence A126311 can accommodate also those cases.
%e A126301 This sequence consists of those terms of A071158 for which 
the first factorial digit is equal to the number of 1's in the term, and 
the following algorithm results the remaining factorial digits of the 
same term: First, extract all maximal subsequences from the term (for d 
ranging from 1 to the largest digit present) that consist of digits d 
and d+1, and place them next to each other, from left to right. E.g. for 
the term 2354543212221 this yields the sequence: 
2212221,2332222,3443,5454,55. After discarding the last digit (here 5), 
and replacing in each batch the smaller number with +1, and larger 
number with -1, we get:
%e A126301 
-1,-1,+1,-1,-1,-1,+1,+1,-1,-1,+1,+1,+1,+1,+1,-1,-1,+1,-1,+1,-1,+1,+1.
%e A126301 and summing these from RIGHT, we get the following partial sums:
%e A126301 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 
1, 2, 1.
%e A126301 Retaining only the partial sums under the +1's (i.e. the 
rightmost one, and all the partial sums that are larger than the 
preceding partial sum one step to the right) we obtain: 
3,5,4,5,4,3,2,1,2,2,2 and 1. These, after appended to the number of 1's 
in the original term (2), yields the same term 2354543212221 from which 
we started from, which thus is a member of this sequence. Similarly, the 
term 2432211 belongs to this sequence, because the same procedure yields:
%e A126301 22211,2322,43,4 and after discarding the last 4:
%e A126301 -1,-1,-1,+1,+1,+1,-1,+1,+1,-1,+1 and summing from the right:
%e A126301 1, 2, 3, 4, 3, 2, 1, 2, 1, 0, 1.
%e A126301 collecting all the partial sums larger than their right 
neighbor (those under +1's), which appended after the number of 1's (2), 
results the same term 2432211.
%Y A126301 a(n) = A071158(A126300(n)) = A007623(A126311(n)). Subset of 
A126299. Cf. A126295.
%K A126301 nonn,hard,more,base
%O A126301 0,3
%A A126301 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 
02 2007


Yours,

Antti

>Hugo
>
>  
>