A003483 -id:A003483 - cp's OEIS frontend

A338575 Number of even permutations on n letters that have a root.

1, 3, 9, 45, 270, 1680, 11340, 108108, 1005480, 10929600, 114845445, 1543647105, 20367547200, 305087041350, 4428663384600, 76921682828760, 1322260935065280, 24987504206574000, 464475474578648925, 9916256376966427425, 209858853988540310400, 4809961676643673161150

Offset: 2

Michel Marcus, Nov 04 2020

nonn

Lev Glebsky, Melany Licón, and Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
M. R. Pournaki, On the number of even permutations with roots, Australasian Journal of Combinatorics, Volume 45 (2009), Pages 37-42. See Theorem 1.1 p. 37.

Cf. A003483.

m = 24;
(Sqrt[(1+x)/(1-x)] Product[Cosh[x^(2k)/(2k)], {k, 1, m}] - (1/2) Product[1 + x^(2k-1)/(2k-1), {k, 1, m}] (Product[Cosh[x^(2k)/(2k)], {k, 1, m}] - Product[Cos[x^(2k)/(2k)], {k, 1, m}]) + O[x]^m // CoefficientList[#, x]&)Range[0, m-1]! // Drop[#, 2]& (* Jean-François Alcover, Nov 17 2020 *)

my(N=30, x='x+O('x^N), v=Vec(serlaplace(sqrt((1+x)/(1-x))*prod(k=1, N, cosh(x^(2*k)/(2*k))) - (1/2)*prod(k=1, N, 1+x^(2*k-1)/(2*k-1))*(prod(k=1, N, cosh(x^(2*k)/(2*k))) - prod(k=1, N, cos(x^(2*k)/(2*k))))))); vector(#v-2, k, v[k+2])

E.g.f.: sqrt((1+x)/(1-x))*Product_{k>=1} cosh(x^(2*k)/(2*k)) - (1/2)*Product_{k>=1} (1+x^(2*k-1)/(2*k-1)) * (Product_{k>=1} cosh(x^(2*k)/(2*k)) - Product_{k>=1} cos(x^(2*k)/(2*k))) for n >= 2.

A373691 Triangle read by rows T(n, k) is the number of permutations on n elements whose square has k descents, for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 10, 6, 6, 2, 26, 22, 48, 22, 2, 76, 68, 276, 260, 40, 0, 232, 214, 1384, 2204, 944, 62, 0, 764, 672, 6240, 16172, 13212, 3048, 200, 12, 2620, 2204, 27096, 103588, 145160, 70740, 10936, 524, 12, 9496, 7354, 113722, 612178, 1338370, 1145614, 364366, 36838, 862, 0

Offset: 1

Michel Marcus, Jun 14 2024

			Triangle begins:
    1;
    2,   0;
    4,   2,    0;
   10,   6,    6,    2;
   26,  22,   48,   22,   2;
   76,  68,  276,  260,  40,  0;
  232, 214, 1384, 2204, 944, 62, 0;
  ...

Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024. See Table 5 p. 13.

Cf. A000085 (1st column), A000142 (row sums), A037224 (right diagonal).

Cf. A003483 (square permutations), A008292.

sq(p) = vector(#p, k, p[p[k]]);
nbd(p) = sum(i=1, #p-1, p[i+1] < p[i]);
row(n) = my(v=vector(n)); for (i=1, n!, v[nbd(sq(numtoperm(n, i)))+1]++;); v;

cp's OEIS Frontend

A338575 Number of even permutations on n letters that have a root.

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