A127378 -id:A127378 - cp's OEIS frontend

A127389 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A127377/A127378 and A127387.

1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716, 14828, 38717, 101682, 268416, 711810, 1895432, 5066030, 13586082, 36547534, 98593064, 266661162, 722953814, 1964358938, 5348367006, 14589803090, 39870312218, 109136843138

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This is INVERTi transform of A086625 (appropriately shifted). I.e. INVERT([1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716]) gives: 1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, ... (beginning of A086625)

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1 + x*A^2)*(1+x)/(1+x+2*x^2));polcoeff(A,n)}

G.f. satisfies: A(x) = (1 + x*A(x)^2)*(1+x)/(1+x+2*x^2).

Generating function, PARI-program and most of the terms supplied by Paul D. Hanna, Jan 15 2007

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.

A127377 Signature-permutation of a Catalan automorphism, auxiliary bijection for Callan's 2006 bijection on Dyck Paths.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 22, 21, 20, 17, 18, 14, 15, 19, 9, 10, 16, 11, 13, 12, 64, 63, 62, 58, 59, 54, 55, 61, 45, 46, 57, 48, 50, 49, 37, 38, 39, 41, 40, 51, 52, 60, 23, 24, 56, 25, 27, 26, 42, 43, 53, 28, 29, 44, 33, 36, 35, 47, 30, 34, 31, 32, 196, 195, 194, 189, 190

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Used to construct A127379.

Inverse: A127378. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127383 and A127389. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, ... A127387 shows a variant which is an involution. A127302(a(n)) = A127302(n) holds for all n.

A127380 Signature-permutation of the inverse of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 13, 12, 14, 15, 19, 21, 22, 16, 20, 18, 17, 23, 24, 25, 27, 26, 28, 29, 33, 35, 36, 30, 34, 32, 31, 37, 38, 39, 41, 40, 51, 52, 56, 58, 59, 60, 62, 64, 63, 42, 43, 53, 54, 55, 47, 61, 49, 50, 44, 57, 48, 46, 45, 65, 66, 67, 69, 68, 70, 71

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This automorphism is RIBS-transformation (explained in A122200) of the automorphism A127378 and Callan's original variant A127382 is obtained as A057164(A127380(A057164(n))).

D. Callan, A Bijection on Dyck Paths and Its Cycle Structure, 2006, 17pp.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A127379. a(n) = A057164(A127382(A057164(n))). A127302(a(n)) = A127302(n) holds for all n.

Differs from A073288 for the first time at n=49, where a(n)=64, while A073288(49)=63 and differs from A122350 for the first time at n=54, where a(n)=54, while A122350(54)=57.

A127387 Signature-permutation of a Catalan automorphism, a self-inverse variant of A127377.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 5, 4, 22, 21, 20, 18, 17, 14, 15, 19, 13, 12, 16, 11, 10, 9, 64, 63, 62, 59, 58, 54, 55, 61, 50, 49, 57, 48, 46, 45, 37, 38, 39, 41, 40, 51, 52, 60, 36, 35, 56, 34, 32, 31, 42, 43, 53, 28, 29, 47, 33, 27, 26, 44, 30, 25, 24, 23, 196, 195, 194, 190, 189

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Used to construct A127388.

The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this involution are given by A127385 and A127389. (This automorphism has the same fixed points as A127377/A127378). A127302(a(n)) = A127302(n) holds for all n.

cp's OEIS Frontend

A127389 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A127377/A127378 and A127387.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

A127377 Signature-permutation of a Catalan automorphism, auxiliary bijection for Callan's 2006 bijection on Dyck Paths.

Views

Author

Keywords

Comments

Links

Crossrefs

A127380 Signature-permutation of the inverse of Callan's 2006 bijection on Dyck Paths, mirrored version (A057164-conjugate).

Views

Author

Keywords

Comments

Links

Crossrefs

A127387 Signature-permutation of a Catalan automorphism, a self-inverse variant of A127377.

Views

Author

Keywords

Comments

Links

Crossrefs