A374993 Total cost when the elements of the n-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the transpose updating strategy.
0, 1, 2, 2, 3, 4, 3, 3, 4, 4, 3, 5, 4, 5, 4, 4, 5, 5, 6, 5, 5, 5, 6, 6, 5, 5, 4, 6, 5, 6, 5, 5, 6, 6, 6, 6, 5, 7, 7, 6, 6, 8, 4, 6, 7, 8, 7, 7, 6, 6, 7, 6, 6, 6, 7, 7, 6, 6, 5, 7, 6, 7, 6, 6, 7, 7, 7, 7, 8, 8, 7, 7, 7, 7, 8, 8, 6, 8, 8, 7, 7, 8, 6, 10, 6, 6, 7
Offset: 0
Keywords
Examples
For n=931 (the smallest n for which A374992(n), A374994(n), A374995(n), and a(n) are all distinct), the 931st composition is (1, 1, 2, 4, 1, 1), giving the following development of the list:
list | position of requested element
--------+------------------------------
1 2 3 4 | 1
^ |
1 2 3 4 | 1
^ |
1 2 3 4 | 2
^ |
2 1 3 4 | 4
^ |
2 1 4 3 | 2
^ |
1 2 4 3 | 1
^ |
---------------------------------------
a(931) = 11
References
- D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, 1998, pp. 401-403.
Links
- John Tyler Rascoe, Table of n, a(n) for n = 0..10000
- Ran Bachrach and Ran El-Yaniv, Online list accessing algorithms and their applications: recent empirical evidence, Proceedings of the 8th annual ACM-SIAM symposium on discrete algorithms, SODA ’97, New Orleans, LA, January 5-7, 1997, 53-62.
- Wikipedia, Self-organizing list.
Crossrefs
Programs
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Python
def comp(n): # see A357625 return def A374993(n): if n<1: return 0 cost,c = 0,comp(n) m = list(range(1,max(c)+1)) for i in c: j = m.index(i) cost += j+1 if j > 0: m.insert(j-1,m.pop(j)) return cost # John Tyler Rascoe, Aug 02 2024
Formula
The sum of a(j) over all j such that A000120(j) = k (number of requests) and A333766(j) <= m (upper bound on the requested elements) equals m^k * k * (m+1)/2. This is a consequence of the fact that the first m positions of the list are occupied by the elements 1, ..., m, as long as no element larger than m has been requested so far.
Comments