A317527 -id:A317527 - cp's OEIS frontend

A369596 Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.

1, 0, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1, 44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1, 265, 44, 44, 53, 53, 62, 64, 29, 22, 24, 16, 16, 8, 6, 5, 4, 2, 1, 1, 0, 0, 1, 1854, 265, 265, 309, 309, 353, 362, 406, 150, 159, 126, 126, 93, 86, 44, 36, 29, 19, 19, 9, 7, 5, 4, 2, 1, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 02 2024

			T(3,0) = 2: 231, 312.
T(3,1) = 1: 132.
T(3,2) = 1: 321.
T(3,3) = 1: 213.
T(3,6) = 1: 123.
T(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Triangle T(n,k) begins:
   1;
   0, 1;
   1, 0, 0,  1;
   2, 1, 1,  1,  0,  0, 1;
   9, 2, 2,  3,  3,  2, 1, 1, 0, 0, 1;
  44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Permutation

Column k=0 gives A000166.
Column k=3 gives A000255(n-2) for n>=2.
Row sums give A000142.
Row lengths give A000124.
Reversed rows converge to A331518.
T(n,n) gives A369796.
Cf. A000217, A001710, A008289, A008290, A038720, A053632, A062869, A124327, A143946, A143947, A161680, A263753, A263756, A317527, A367955, A368338.

b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
      `if`(j=n, x^j, 1)*b(s minus {j})), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7);
# second Maple program:
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
    end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);

g[n_] := g[n] = If[n == 0, 1, n*g[n - 1] + (-1)^n];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
   If[n == 0, g[m], b[n, i-1, m] + b[n-i, Min[n-i, i-1], m-1]]];
T[n_, k_] := b[k, Min[n, k], n];
Table[Table[T[n, k], {k, 0, n*(n + 1)/2}], {n, 0, 7}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)

Sum_{k=0..A000217(n)} k * T(n,k) = A001710(n+1) for n >= 1.
Sum_{k=0..A000217(n)} (1+k) * T(n,k) = A038720(n) for n >= 1.
Sum_{k=0..A000217(n)} (n*(n+1)/2-k) * T(n,k) = A317527(n+1).
T(n,A161680(n)) = A331518(n).
T(n,A000217(n)) = 1.

A181967 Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.

Original entry on oeis.org

0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2634508800, 37362124800, 566658892800, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 25545471085854720000, 587545834974658560000, 13488008733331292160000

Offset: 1

Olivier Gérard, Apr 04 2012

nonn

The first 11 terms of this sequence are the same as A317527. - Andrew Howroyd, Jul 30 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A181951 for the number of such subgroups.
Cf. A181966 is the symmetric group case.
Cf. A013939, A317527.

List([1..7], n->Sum(Filtered( ConjugacyClassesSubgroups( AlternatingGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( AlternatingGroup(n), Representative(x))) )); # Andrew Howroyd, Jul 30 2018

a:=function(n) local total, perm, g, p, k;
  total:= 0; g:= AlternatingGroup(n);
  for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
     if p>2 or IsEvenInt(k) then
       perm:=PermList(List([0..p*k-1], i->i - (i mod p) + ((i + 1) mod p) + 1));
       total:=total + Size(Normalizer(g, perm)) * Factorial(n) / (p^k * (p-1) * Factorial(k) * Factorial(n-k*p));
     fi;
  od; od;
  return total;
end; # Andrew Howroyd, Jul 30 2018

a(n)={n!*sum(p=2, n, if(isprime(p), if(p==2, n\4, n\p)))/2} \\ Andrew Howroyd, Jul 30 2018

a(n) = n! * (A013939(n) - floor((n + 2)/4)) / 2. - Andrew Howroyd, Jul 30 2018

Some incorrect conjectures removed by Andrew Howroyd, Jul 30 2018

Terms a(9) and beyond from Andrew Howroyd, Jul 30 2018

A308599 Number of (not necessarily maximal) cliques in the n-alternating group graph.

Original entry on oeis.org

2, 8, 45, 301, 2281, 19321, 181441, 1874881, 21168001, 259459201, 3432844801, 48778329601, 741015475201, 11987015040001, 205740767232001, 3734717995008001, 71493173047296001, 1439467021504512001, 30411275102208000001, 672697405260840960001, 15548676734256906240001

Offset: 2

Eric W. Weisstein, Jun 09 2019

nonn

The maximum size of a clique in the n-alternating graph is 3. Cliques are then triangles, edges, vertices and the empty set. - Andrew Howroyd, Sep 08 2019

Andrew Howroyd, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Alternating Group Graph
Eric Weisstein's World of Mathematics, Clique

Cf. A001710, A317527.

a(n) = 1 + (4*n - 5)*n!/6; \\ Andrew Howroyd, Sep 08 2019

a(n) = 1 + (4*n - 5)*n!/6. - Andrew Howroyd, Sep 08 2019

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Sep 08 2019

cp's OEIS Frontend

A369596 Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.

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A181967 Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.

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A308599 Number of (not necessarily maximal) cliques in the n-alternating group graph.

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PARI

Formula

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