A014138 -id:A014138 - cp's OEIS frontend

A089402 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089864.

1, 1, 2, 3, 8, 22, 68, 217, 720, 2438, 8412, 29414, 104048, 371516, 1337352, 4847637, 17679264, 64823110, 238820780, 883634026, 3282065072, 12233141908, 45741298616, 171529836218, 644952132448, 2430973304732, 9183676744088

Offset: 0

Antti Karttunen, Nov 29 2003

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

A. Karttunen, C-program for computing the initial terms of this sequence

(define (A089402 n) (* (/ 1 2) (+ (A000108 n) (A089408 n))))

a(n) = (A000108(n)+A089408(n))/2

A089407 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089859/A089863.

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 34, 109, 360, 1219, 4206, 14708, 52024, 185758, 668676, 2423821, 8839632, 32411555, 119410390, 441817020, 1641032536, 6116570954, 22870649308, 85764918130, 322476066224, 1215486652366, 4591838372044, 17383387914792

Offset: 0

Antti Karttunen, Nov 29 2003

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

A. Karttunen, C-program for computing the initial terms of this sequence

(define (A089407 n) (if (< n 2) 1 (* (/ 1 4) (+ (A000108 n) (A089408 n) (if (= 3 (modulo n 4)) (* 2 (A000108 (/ (- n 3) 4))) 0)))))

a(0) = a(1) = 1, a(n) = (1/4)*(A000108(n) + A089408(n) + (if (n == 3 mod 4) 2*A000108((n-3)/4)))

A089412 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A074683/A074684.

Original entry on oeis.org

1, 1, 2, 5, 18, 84, 2793, 211123440, 140826255570, 213340617315, 156232599082560

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

A. Karttunen, C-program for computing the initial terms of this sequence

A089413 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071655/A071656.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 5, 5, 11, 11, 17, 11, 21, 17, 30, 22, 52, 24, 74, 36

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

A. Karttunen, C-program for computing the initial terms of this sequence

Bisections: A089409, A089416. Cf. A089411.

A127278 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A126313/A126314.

Original entry on oeis.org

1, 1, 0, 2, 4, 2, 0, 4, 0, 1, 0, 0, 0

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Fixed points themselves are given in A127306. Cf. A127282.

A127279 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A126313/A126314.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 10, 32, 79, 176, 612, 644, 8547

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

A127283.

A127383 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127377/A127378.

Original entry on oeis.org

1, 1, 1, 2, 6, 15, 46, 141, 446, 1427, 4722, 15884, 54224, 187380

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

A127384, A127385, A127389.

A130918 Simple self-inverse permutation of natural numbers: List each block of A000108(n) numbers from A014137(n-1) to A014138(n-1) in reverse order.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 5, 4, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 196, 195, 194, 193, 192

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

In principle this involution is the signature permutation of yet another Catalan automorphism. However, the question remains what is the most "natural" way to create such an automorphism acting e.g. on S-expressions (i.e. rooted plane binary trees), which would produce this sequence as its signature permutation.

A. Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Signature permutations of Catalan bijections
Index entries for signature-permutations of Catalan automorphisms

Inverse: A130918. Cf. A054429, A057163. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A130380 and A036987.

(define (A130918 n) (if (zero? n) n (- (A014138 (- (A072643 n) 1)) (A082853 n))))

a(0)=0, a(n) = A014138(A072643(n)-1) - A082853(n).

A081152 Number of even cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506.

Original entry on oeis.org

0, 1, 6, 37, 212, 1226, 6822, 38081, 215414

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Cf. A081151.
Cf. A081150, A081159, A081161, A081163.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081150(2n+1) = A081159(n)+A081161(n)+A081163(n).

A081156 Number of odd cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506, with fixed-points of both A057163 and A057164.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 20, 59, 120

Offset: 0

Wouter Meeussen & Antti Karttunen Mar 10 2003

See comment at A080973.

Cf. A080973.
Cf. A081155, A000108, A081163.
Cf. A014137, A014138, A057505, A057506, A057163, A057164.

a(n) = A081155(2n+1) = A000108(n) - 2*A081163(n).

cp's OEIS Frontend

A089402 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089864.

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A089407 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089859/A089863.

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A089412 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A074683/A074684.

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A089413 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071655/A071656.

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A127278 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A126313/A126314.

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A127279 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A126313/A126314.

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A127383 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127377/A127378.

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A130918 Simple self-inverse permutation of natural numbers: List each block of A000108(n) numbers from A014137(n-1) to A014138(n-1) in reverse order.

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A081152 Number of even cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506.

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A081156 Number of odd cycles in range [A014137(2n)..A014138(2n)] of permutation A057505/A057506, with fixed-points of both A057163 and A057164.

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