

A127301


MatulaGoebel signatures for plane general trees encoded by A014486.


14



1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36
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OFFSET

0,2


COMMENTS

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n1)..A014138(n)] of A014486 to A000081(n+1) distinct nonoriented rooted general trees, encoded by their MatulaGoebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signaturepermutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the nonoriented form of a general tree, which implies also that it is Łukasiewiczword permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence.  Antti Karttunen, Jan 03 2013


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to MatulaGoebel numbers


FORMULA

A001222(a(n)) = A057515(n) for all n.


EXAMPLE

A000081(n+1) distinct values occur each range [A014137(n1)..A014138(n1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....
.o...o
..\./.
...*..
MatulaGoebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.
.o...o
..\./.
...*..
MatulaGoebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.


PROG

(Scheme:) (define (A127301 n) (*A127301 (A014486>parenthesization (A014486 n)))) ;; A014486>parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (foldleft (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))


CROSSREFS

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(A106191(n)) = A033844(n1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
Sequence in context: A246166 A264802 A243493 * A209636 A243491 A271863
Adjacent sequences: A127298 A127299 A127300 * A127302 A127303 A127304


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jan 16 2007


STATUS

approved



