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A242615 Number of messages maximally transmittable by using n objects as a non-sequenced collection, expressed as a percentage of the number of messages maximally transmittable by using n objects as a sequenced collection. 1
100, 81, 56, 56, 37, 44, 30, 35, 29, 32, 27, 33, 29, 32, 31, 35, 34, 39, 38, 43, 44, 48, 50, 56, 58, 64, 67, 74, 77, 85, 90, 98, 104, 113, 119, 130, 137, 147, 156, 167, 176, 188, 197, 210, 220, 232, 243, 255, 265, 278, 288, 299, 309, 320, 328, 338, 345, 354, 360, 367, 371, 376, 378, 381, 382, 383, 381, 380, 377, 373, 368, 363, 356, 349, 341, 332, 322, 312, 302, 291, 279, 268, 256, 244, 232, 220, 208, 196, 184, 173, 161, 151, 140, 130, 120, 111, 102, 94, 86, 78, 71, 65, 59, 53, 48, 43, 38, 34, 31, 27, 24, 21, 19, 17, 15, 13, 11, 10, 8, 7, 6, 5, 5, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A242615 compares the number of distinct (distinguishable) states a collection can be in. There are two ways to read symbols off a collection: the sequential and the commutative ways. In an idealized case, the objects seen as sequentially ordered will each possess one symbol which individuates this element fully against all other elements in the collection. Then, there are n! distinct states of the collection achievable. This is also the maximal number of messages that can be transmitted by using n objects as elements in a sequence.

Reading the collection by building groups among elements that share a symbol one arrives at an upper limit, here called for ease of comparison "n?", which is given by the number of partitions of n raised to the power of the logarithm of the number of partitions of n.

Being sequenced or non-sequenced is a property that the human spectator looks into the collection (cf. Rorschach-Test). The collection itself possesses the immanent, intrinsic property of being both sequenced and non-sequenced.

The discussion centers on what is known in data processing as sequential vs. index-based retrieval. We investigate how many index queries are necessary until one identifies each one specific element of a data set; would this method be more efficient than the usual method of assigning each element a sequential number.

The numbers show that a translation exists between group and sequential properties. The non-integer result for n? comes from the decreasing probability of successive index searches to bring forth such elements that have not yet been found in the course of previous index searches.

The interplay between linear and nonlinear order concepts can be used efficiently and practically by making use of the chains that connect elements into groups (thus creating commutative, contemporary group relations) and at the same time assigning to each element a sequential number that refers to the place within the chains' succession. (Cf. A235647 for a definition of chains.)

LINKS

Table of n, a(n) for n=1..150.

Karl Javorszky, Graph of n?/n!

Karl Javorszky, Transfer of Genetic Information: An Innovative Model, Proceedings 2017, 1, 222.

Karl Javorszky, "Data versus information", Krassimir Markov, ed., Information Theories and Applications (2017), Vol. 24, No. 4, see pages 308 & 317.

Karl Javorszky, Picturing Order, Contemporary Computational Science (2018), 3rd Conf. on Inf. Tech. Systems Res. and Comp. Phys. (ITSRCP18), 83-91.

FORMULA

A242615 = exp( log(A000041)^2 - log(A000142) ) * 100.

EXAMPLE

For n=1, both n? and n! yield 1; near n=32 and n=97 n? ~ n!, for n > 136, the two functions diverge.

CROSSREFS

Cf. A000041, A000142, A235647.

Sequence in context: A234322 A260707 A180102 * A090292 A020993 A115020

Adjacent sequences:  A242612 A242613 A242614 * A242616 A242617 A242618

KEYWORD

nonn

AUTHOR

Karl Javorszky, May 19 2014

STATUS

approved

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Last modified January 29 07:03 EST 2020. Contains 331337 sequences. (Running on oeis4.)