Chad Brewbaker - cp's OEIS frontend

A285053 Multiplications between idempotent equivalence classes for n X n matrices over GF(2).

1, 4, 118, 27080

Offset: 1

Chad Brewbaker, Apr 08 2017

With the n X n matrices over GF(2) construct 3-tuples (a,b,c) where a*b = c. Map each of the three elements to their idempotent under self multiplication. Filter on unique 3-tuples.

The idempotents are enumerated in A132186.

A285051 Idempotent equivalence class multiplications for the full transformation semigroup.

Original entry on oeis.org

1, 11, 268, 13705, 1122726

Offset: 1

Chad Brewbaker, Apr 08 2017

Over all n^n transformations create a list of 3-tuples (a,b,c) where a*b = c. For each tuple map each of the three elements to its idempotent under self-composition, (g^i)*(g^i) = g^i. Collect the unique 3-tuples.

			For n=2: [([0,0],[0,0],[0,0]),([0,0],[0,1],[0,0]),([0,0],[1,1],[0,0]),([0,1],[0,0],[0,0]),([0,1],[0,1],[0,1]),([0,1],[1,1],[1,1]),([0,1],[0,0],[1,1]),([0,1],[1,1],[0,0]),([1,1],[0,0],[1,1]),([1,1],[0,1],[1,1]),([1,1],[1,1],[1,1])]

A285055 Sum of the cycle lengths of the elements 0,1,...n-1, in Zn under multiplication.

Original entry on oeis.org

1, 2, 4, 5, 12, 8, 22, 11, 24, 24, 64, 18, 78, 44, 38, 31, 172, 48, 184, 50, 74, 128, 334, 38, 236, 156, 192, 96, 474, 76, 442, 103, 214, 344, 194, 106, 672, 368, 242, 102, 904, 148, 904, 278, 218, 668, 1522, 102, 910, 472, 518, 320, 1728, 384, 558, 200, 614, 948

Offset: 1

Chad Brewbaker, Apr 08 2017

nonn

			For n=3, 0->0 (cycle length 1), 1->1 (cycle length 1) and 2->1->2 (cycle length 2); 1+1+2 = 4.
The triangle of the cycle lengths is:
1 ;
1, 1 ;
1, 1, 2 ;
1, 1, 1, 2 ;
1, 1, 4, 4, 2 ;
1, 1, 2, 1, 1, 2 ;
1, 1, 3, 6, 3, 6, 2 ;
1, 1, 1, 2, 1, 2, 1, 2 ;
1, 1, 6, 1, 3, 6, 1, 3, 2 ;
1, 1, 4, 4, 2, 1, 1, 4, 4, 2 ;

See A060014 for permutations on n elements.

A285055Tr := proc(n,m)
    local a,nmap,cyc ;
    nmap := n ;
    cyc := [n] ;
    for a from 1 do
        nmap := modp(nmap*n,m) ;
        if member(nmap,cyc,'i') then
            return nops(cyc)-i+1;
        else
            cyc := [op(cyc),nmap] ;
        end if;
    end do:
end proc:
A285055 := proc(m)
    add(A285055Tr(n,m),n=0..m-1) ;
end proc:
seq(A285055(m),m=1..30) ; # R. J. Mathar, May 06 2017

A285052 Number of idempotent equivalence classes for multiplication in Zn.

Original entry on oeis.org

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 4, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 4, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 16, 4, 16, 16, 16, 4, 16, 16, 16, 16, 16, 4, 64, 4, 16, 16, 4, 16, 64, 4, 16, 16, 64, 4, 16, 4, 16, 16, 16, 16, 64, 4, 16, 4, 16, 4, 64, 16, 16, 16, 16, 4, 64, 16

Offset: 1

Chad Brewbaker, Apr 08 2017

nonn

Consider triples (a,b,c) over Zn where a*b=c. Map each of the three elements to its idempotent under self multiplication, (g^i) * (g^i) = (g^i). Count the distinct triples.

			For n=6: [(0,0,0),(0,1,0),(0,4,0),(0,3,0),(1,0,0),(1,1,1),(1,4,4),(1,3,3),(4,0,0),(4,1,4),(4,4,4),(4,3,0),(3,0,0),(3,1,3),(3,4,0),(3,3,3)], so a(6) = 16.

Chad Brewbaker, Endoscope: A toolkit for analysis of endofunctions on small sets

Conjecture: a(n) = 4^A001221(n).

A239785 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(g(x))) = f(g(x)).

Original entry on oeis.org

1, 1, 14, 411, 19912, 1412745, 136537056, 17121680443

Offset: 0

Chad Brewbaker, Mar 26 2014

Cf. A181162.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

A239784 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(g(x))) = f(g(f(x))).

Original entry on oeis.org

1, 1, 14, 351, 14776, 902605, 75313056, 8194699387

Offset: 0

Chad Brewbaker, Mar 26 2014

Cf. A181162.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

A239783 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(x)) = f(g(f(x))).

Original entry on oeis.org

1, 1, 14, 339, 12496, 639005, 42829536, 3624387739

Offset: 0

Chad Brewbaker, Mar 26 2014

Cf. A181162.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

A239782 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(f(x))) = f(f(g(x))).

Original entry on oeis.org

1, 1, 14, 321, 12256, 682985, 52551216, 5333977789

Offset: 0

Chad Brewbaker, Mar 26 2014

Cf. A181162.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

A239768 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(g(f(x))).

Original entry on oeis.org

1, 1, 10, 195, 6808, 362745, 26848656, 2621263519, 324981308800, 49669569764433, 9146879704748800, 1993011341241988551, 506190915590699695104, 148000190814308473203433, 49289886405448749446514688, 18529196186934893511062427375, 7800708229072237749055062900736, 3652486190893312491910941333813537

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..251

Cf. A181162.

A[n_] := If[n == 0, 1, Sum[(n!/(n - k)!) Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
Table[A[n],{n,10}] (* David Einstein, Oct 10 2016 *)

a(n) = sum(k= 0, n, (n!/(n-k)!)*binomial(n,k)*(n*k)^(n-k)); \\ Michel Marcus, Oct 11 2016; corrected Jun 13 2022

a(n) = Sum_{k=0..n} (n!/(n-k)!) * C(n,k) * (n*k)^(n-k). - David Einstein, Oct 10 2016

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(17) from David Einstein, Oct 10 2016

A239766 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = f(g(g(x))).

Original entry on oeis.org

1, 1, 10, 195, 5992, 260085, 14922576, 1087100371

Offset: 0

Chad Brewbaker, Mar 26 2014

Cf. A181162.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

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User: Chad Brewbaker

A285053 Multiplications between idempotent equivalence classes for n X n matrices over GF(2).

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A285051 Idempotent equivalence class multiplications for the full transformation semigroup.

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A285055 Sum of the cycle lengths of the elements 0,1,...n-1, in Zn under multiplication.

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Maple

A285052 Number of idempotent equivalence classes for multiplication in Zn.

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A239785 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(g(x))) = f(g(x)).

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A239784 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(g(x))) = f(g(f(x))).

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A239783 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(x)) = f(g(f(x))).

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Extensions

A239782 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(f(x))) = f(f(g(x))).

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Author

Keywords

Crossrefs

Extensions

A239768 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(g(f(x))).

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Mathematica

PARI

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Extensions

A239766 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = f(g(g(x))).

Views

Author

Keywords

Crossrefs

Extensions