Skolem's Sequences
Introduction
Thoralf Albert Skolem (1787 - 1963), Norwegian mathematician worked in algebra, number theory and logic. He is the founder of the theory of the models.
Theorem of Löwenheim-Skolem: every first order noncontradictory axiomatic admits a countable model. (The paradox of Skolem because this countable model contains uncountable sets).
Theorem of Löwenheim-Skolem: every first order noncontradictory axiomatic admits a countable model. (The paradox of Skolem because this countable model contains uncountable sets).
Definition
Observe this sequence
Did you see the characteristics of these lists? Look at how both are placed
When the positions of a = 1 or a = 2, 3, 4, ... are x (the smallest) and y (the largest), then y = x+a (or y-x = a).
The distance from one a to the other a is a.
4 5 1 1 4 3 5 2 3 2
or this other one 3 5 6 3 8 4 5 7 6 4 1 1 8 2 7 2.
Did you see the characteristics of these lists? Look at how both are placed
1, then look at the positions of 2 ...
When the positions of a = 1 or a = 2, 3, 4, ... are x (the smallest) and y (the largest), then y = x+a (or y-x = a).
The distance from one a to the other a is a.
Existence
It is shown that it can exist of sequence of Skolem only if n or n-1 are multiple of 4.
It is shown that there are series Skolem for all positive integer of the form n = 4p and n = 4p+1
It is shown that there are series Skolem for all positive integer of the form n = 4p and n = 4p+1
Constructions
Application
For n null the sequence is empty. You can build Skolem's sequence for the following values
n=1,
n=4,
n=5,
n=8,
n=9,
n=12,
n=13,
n=16,
n=17,
n=20,
n=21,
n=24,
n=25,
n=28,
n=29, etc.
The number of calculated solutions is voluntarily limited. Click on [Search] for possibly others. Use the buttons [ |<< ] [ <-- ] [ --> ] [ >>| ] to navigate from one solution to another.
The number of calculated solutions is voluntarily limited. Click on [Search] for possibly others. Use the buttons [ |<< ] [ <-- ] [ --> ] [ >>| ] to navigate from one solution to another.
Diagrams
The pairs are represented by segments.
Click to display or hide the diagram
Skolem triple system
Click to display or hide the system of SkolemDifferences system
Click to display or hide the system of differencesSteiner triple system
Introduced for the first time in 1847 by Kirkman (Young ladys diaries),
these systems were rediscovered and published in 1853 by Steiner in a geometrical context. A proof of existence was given by Kirkman (for sets of v=6p+1 and v=6p+3 elements) but not by Steiner.
The problem was posed by Kirkman as follows:
Fifteen young ladies in a school walk out thee abreast for seven days in succession: it is required tho arrange them daily, so that no two shall walk twice abreast.
A block design is a collection of subsets of K elements of a unit S of v elements such as each pair of elements of S appears in exactly lambda blocks. A triple system of Steiner is (v, K, lambda) a design, one (v, K, lambda)BIBD or Steiner 2-design. A block design is a collection of k-subsets of a set S of v elements such as each pair of elements of S appears in exactly lambda blocks. A Steiner triple system is a
design, a 
or a Steiner 2-design.
(Kirkman asks for a resolvable triple system, this constraint implies the form v=6p+3 not verified by the system on this page for which v=6p+1).
The problem was posed by Kirkman as follows:
Fifteen young ladies in a school walk out thee abreast for seven days in succession: it is required tho arrange them daily, so that no two shall walk twice abreast.
A block design is a collection of subsets of K elements of a unit S of v elements such as each pair of elements of S appears in exactly lambda blocks. A triple system of Steiner is (v, K, lambda) a design, one (v, K, lambda)BIBD or Steiner 2-design. A block design is a collection of k-subsets of a set S of v elements such as each pair of elements of S appears in exactly lambda blocks. A Steiner triple system is a
design, a or a Steiner 2-design.
(Kirkman asks for a resolvable triple system, this constraint implies the form v=6p+3 not verified by the system on this page for which v=6p+1).
Click to display or hide the system of Steiner
Number of Skolem sequences
For n > 0, the number of Skolem sequences of order n are
Generally, one do not distinguish a Skolem sequence and the symmetric sequence. In this case, the numbers of Skolem sequences are
(By computer, may/june 2007)
1,0,0,6,10,0,0,504,2636,0,0,455936,3040560 ...
Generally, one do not distinguish a Skolem sequence and the symmetric sequence. In this case, the numbers of Skolem sequences are
1, 0, 0, 3, 5, 0, 0, 252, 1318, 0, 0, 227968, 1520280 ...
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ... |
| Sequences | 1 | 0 | 0 | 3 | 5 | 0 | 0 | 252 | 1318 | 0 | 0 | 227968 | 1520280 | ... |
Algorithms
In the future.
Books, documents, links...
Combinatorial Designs and Tournaments Ian Anderson, Department of Mathematics, University of Glasgow
Des Mathématiciens de A à Z de Bertrand Hauchecorne et Daniel Surreau - ellipses
Encyclopaedia of DesignTheory Bibliography
The electronic journal of combinatorics
Le jeu des cavaliers Académie de VersaillesJean Brette, du Palais de la Découverte, a conçu à partir de ce problème un jeu, le jeu des cavaliers : pour un entier n donné, chaque paire d'entiers identiques p est représentée par un cavalier. Des trous sont ménagés dans une planchette. Le jeu consiste à y ficher les cavaliers de telle sorte que deux trous seulement sur chaque ligne et un seul sur chaque colonne soient occupés.
Sculpture à l'IHÉS l'Institut possède une représentation d'une des suites de Skolem sous forme de sculpture, réalisée par l'artiste américaine Jessica Stockholder, suivant un cahier des charges établi par des classes de primaire qui ont travaillé sur le jeu des cavaliers.
SKOLEM Thoralf Albert, norvégien, 1887-1963 Serge Mehl - www.chronomath.com
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