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A335725 The number of sigma matrices on the set of all endofunctions as a function of domain size n. 0
1, 2, 5, 13, 35, 93, 260 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The number of unique sigma matrices of endofunctions as a function of n where n is the size of the finite domain. The sigma matrix is an n X n preimage data structure in which an arbitrary entry is given by sigma[i,j] = abs(f(x_{i})^{-j}). In other words, given an endofunction on X, the sigma matrix captures the size of the j-back inverse applied to the i-th domain element of X.

REFERENCES

Fournier-Eaton, Bradford M., "A Theory of Preimage Complexity: Data-structures, Complexity Measures and Applications to Endofunctions and Associated Digraphs" (2020). University of New Orleans Theses and Dissertations. 2794.

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

A two element domain corresponds to n=2. There are 2^2=4 endofunctions on two elements. However the only unique sigma matrices correspond to S1 = [[2,2],[0,0]] and S2 = [[1,1],[1,1]], and thus sigma(2)=2. See the referenced dissertation at the associated link for a full exposition including examples, definitions and theory.

CROSSREFS

Sequence in context: A291242 A097919 A160438 * A240609 A054657 A024576

Adjacent sequences:  A335722 A335723 A335724 * A335726 A335727 A335728

KEYWORD

nonn,hard,more

AUTHOR

Bradford M. Fournier-Eaton, Jun 19 2020

STATUS

approved

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Last modified October 16 09:02 EDT 2021. Contains 348041 sequences. (Running on oeis4.)