A303979 Triangle read by rows: T(n,k) is the number of cyclic unimodal permutations of length n with a peak at position k.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 4, 4, 3, 1, 0, 0, 1, 3, 6, 8, 6, 3, 1, 0, 0, 1, 3, 9, 13, 12, 8, 4, 1, 0, 0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0, 0, 1, 5, 13, 27, 39, 39, 27, 13, 5, 1, 0

Offset

1,19

Links

Table of n, a(n) for n=1..78.

Formula

T(n,k) = Sum_{j=1..k-1} (-1)^(k+j+1)*A051168(n,j), when n is odd and n>2;

T(n,k) = Sum_{j=1..k-1} (-1)^(k+j+1)*A051168(n,j)+(-1)^(k+1)*Sum_{j<k, n-j==2 (mod 4)} A051168(n/2, j/2), when n is even and n>2.

Example

For n = 5, there are 6 unimodal cyclic permutations: 234561, 235641, 246531, 345621, 465321. There are T(6,1) = 0 with peak at position 1, T(6,2) = 1 with peak at position 2, T(6,3) = 1 with peak at position 3, T(6,4) = 2 with peak at position 4, T(6,5) = 1 with peak at position 5, and T(6,6) = 0 with peak at position 6.
Starting at n=1 with 1 <= k <= n, the triangle begins:
  0,
  0, 0,
  0, 1, 0,
  0, 1, 1, 0,
  0, 1, 1, 1, 0,
  0, 1, 1, 2, 1, 0,
  0, 1, 2, 3, 2, 1, 0,

Prog

(PARI) t051168(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));
T(n, k) = my(t=sum(j=1, k-1, (-1)^(k+j+1)*t051168(n, j))); if (!(n % 2), t += (-1)^(k+1)*sum(j=1, k-1, if (((n-j) % 4) == 2, t051168(n/2, j/2)))); t;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 16 2018

Crossrefs

Cf. A051168.

Sequence in context: A025880 A058755 A128519 * A301573 A061670 A236412

Adjacent sequences: A303976 A303977 A303978 * A303980 A303981 A303982

Keyword

nonn,tabl

Author

Kassie Archer, May 03 2018

Status

approved