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

Author

Kassie Archer, May 03 2018

Keywords

• nonn
• tabl

Examples

			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,
		

Crossrefs

Cf. A051168.

Programs

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

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_{jA051168(n/2, j/2), when n is even and n>2.