A303979 -id:A303979 - cp's OEIS frontend

A303980 a(n) is the number of cyclic permutations that admit a [1,1,-1]-gridding.

1, 1, 1, 2, 5, 15, 42, 120, 338, 952, 2671, 7494, 21035, 59115, 166432, 469560, 1327802, 3763545, 10692500, 30447858, 86894361, 248506757, 712109662, 2044402512, 5879579540, 16937048040, 48864612667, 141179970820, 408444645375, 1183143522435, 3431241484224, 9961919944284

Offset: 0

Kassie Archer, May 03 2018

nonn

a(n) is the number of cyclic permutations that, when written in their one-line notation, is composed of an increasing segment, followed by another increasing segment, followed by a decreasing segment.

Cf. A027376, A303979, A006575.

t051168(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));
T303979(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;
a027376(n) = if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n);
a006575(n) = sumdiv( n, d, if ( bitand(d, 1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n);
a(n) = if (n <= 2, 1, res = a027376(n)/2 - sum(i=2, n-1, i*T303979(n,i)); if (!(n%2), res += a006575(n/2)/2); res); \\ Michel Marcus, May 16 2018

a(n) = A027376(n)/2 - Sum_{i=2..n-1} i*A303979(n,i) when n is odd and n>2.

a(n) = (A027376(n)+A006575(n/2))/2 - Sum_{i=2..n-1} i*A303979(n,i) when n is even and n>2.

More terms from Michel Marcus, May 16 2018

A304201 a(n) is the number of cyclic permutations of length n that admit a [1,-1,-1]-gridding.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 43, 120, 338, 952, 2672, 7494, 21035, 59115, 166433, 469560, 1327802, 3763545, 10692500, 30447858, 86894361, 248506757, 712109663, 2044402512, 5879579540, 16937048040, 48864612668, 141179970820, 408444645375, 1183143522435, 3431241484223, 9961919944284

Offset: 0

Kassie Archer, May 08 2018

nonn

a(n) is the number of permutations of length n that are composed of an increasing segment, followed by a decreasing segment, followed by another decreasing segment. In other words, these permutation have a descent set of the form {i, i+1, ..., n-1} for some i or {i, i+1, ..., n-1}\{j} for some i and j > i.

Cf. A027376, A303979, A133267.

t051168(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));
T303979(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;
a027376(n) = if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n);
a133267(n) = sumdiv(n, d, moebius(d)*3^(n/d)/n - if (d%2, moebius(d)*(3^(n/d)-1)/(2*n)));
a006575(n) = sumdiv(n, d, if ( bitand(d, 1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n);
a(n) = if (n <= 2, 1, res = a027376(n)/2 - sum(i=2, n-1, (n+1-i)*T303979(n,i)); if (!(n%2), if ((n%4)==2, res += a133267(n/2)/2, res += a006575(n/2)/2)); res); \\ Michel Marcus, May 18 2018

a(n) = A027376(n)/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n is odd and n > 2;
a(n) = (A027376(n) + A133267(n/2))/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n = 2 (mod 4) and n > 2.
a(n) = (A027376(n) + A006575(n/2))/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n = 0 (mod 4) and n > 2.

More terms from Michel Marcus, May 19 2018

cp's OEIS Frontend

A303980 a(n) is the number of cyclic permutations that admit a [1,1,-1]-gridding.

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