cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A125988 Signature-permutation of the square of A125986.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 16, 12, 17, 21, 10, 13, 11, 14, 20, 18, 15, 19, 22, 23, 42, 30, 45, 56, 26, 35, 31, 44, 58, 48, 46, 57, 63, 24, 32, 27, 33, 50, 25, 29, 28, 37, 53, 47, 40, 54, 62, 43, 36, 34, 38, 41, 49, 39, 51, 61, 59, 55, 52, 60, 64, 65, 121, 84, 128, 163, 72
Offset: 0

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Author

Antti Karttunen, Jan 02 2007

Keywords

Links

Crossrefs

Inverse: A125987. a(n) = A125986(A125986(n)).

A126300 Fixed points of the permutation A125985/A125986.

Original entry on oeis.org

0, 1, 47, 443, 653479, 10269977
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

Those i, for which A125985(i)=i. A126301 shows the same fixed points using factorial code as employed in Vaillé's paper.

Links

Crossrefs

Subset of A126298. Cf. A126295. A126311(n) = A071156(a(n)).

A126291 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A125985/A125986.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 8, 11, 16, 23, 22, 32, 44, 54
Offset: 0

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Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

The number of orbits (equivalence classes) to which Vaille's automorphism partitions the set of A000108(n) Dyck paths of semilength n. Note the non-monotone drop from a(9) to a(10).

Crossrefs

A126294(2n) = 2*a(2n) for n>0.

A126292 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A125985/A125986.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 126, 446, 1438, 9732, 25832, 102990, 306732
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

The size of a largest orbit to which Vaille's automorphism partitions the set of A000108(n) Dyck paths of semilength n.

Crossrefs

For n>0, it seems that a(2n) = 2*A126296(2n).

A126293 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A125985/A125986.

Original entry on oeis.org

1, 1, 2, 6, 12, 780, 27720, 47785500, 160430026680, 19702100764977190560, 1389717843080061549600, 689103351818617666941410400, 3910098498311750671529113672239956967773312909280
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Crossrefs

Cf. A014137, A014138, A125985, A125986, A126297.

A126311 A071156-codes for the fixed points of the permutation A125985/A125986.

Original entry on oeis.org

0, 1, 327, 13383, 14107138025, 4217868316383
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

A126301 shows the same terms in factorial base notation.

Links

Crossrefs

Cf. A071156, A126300.

Formula

a(n) = A071156(A126300(n))

A125985 Signature-permutation of Vaillé's 1997 bijection on 'bridges' (Dyck paths).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

Vaillé shows in 1997 paper that this automorphism transforms a 'derivation' of a Dyck path to its 'compression', i.e., in OEIS terms, A125985(A126310(n)) = A126309(A125985(n)) holds for all n. He also proves that A057515(A125985(n)) = A126307(n) and A057514(A125985(n)) = A072643(n) - A057514(n) + 1 (the latter identity for all n >= 1).

Links

Crossrefs

Inverse: A125986. The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A126291, A126292 and A126293. The fixed points are given by A126300/A126301.

Programs

  • Scheme
    (define (A125985 n) (A080300 (rising-list->binexp (A125985-aux2 (A014486 n)))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (A125985-aux1 n) (if (zero? n) (list) (let ((begin_from (<< 1 (- (- (A000523 n) (A090996 n)) 1)))) (let loop ((s (A090996 n)) (t 0) (nth_list 1) (p begin_from) (b (if (= 0 (A004198bi n begin_from)) 0 1)) (lists (list (list)))) (cond ((< s 1) (cond ((< p 1) (reverse! lists)) (else (loop (- t (- 1 b)) b (+ 1 nth_list) (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (list (+ b 1 nth_list)) lists))))) (else (loop (- s (- 1 b)) (+ t b) nth_list (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (cons (+ b nth_list) (car lists)) (cdr lists)))))))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (m-join a b m) (let loop ((a a) (b b) (c (list))) (cond ((and (not (pair? a)) (not (pair? b))) (reverse! c)) ((not (pair? a)) (loop a (cdr b) (cons (car b) c))) ((not (pair? b)) (loop (cdr a) b (cons (car a) c))) ((equal? (car a) (car b)) (loop (cdr a) (cdr b) (cons (car a) c))) ((> (car b) m) (loop a (cdr b) (cons (car b) c))) (else (loop (cdr a) b (cons (car a) c))))))
(define (rising-list->binexp rising-list) (let loop ((s 0) (i 0) (h 0) (fs rising-list)) (cond ((null? fs) (+ s (<< (- (<< 1 h) 1) i))) ((> (car fs) h) (loop s (+ i 1) (car fs) (cdr fs))) (else (loop (+ s (<< (- (<< 1 (+ 1 (- h (car fs)))) 1) i)) (+ i 2 (- h (car fs))) (car fs) (cdr fs))))))
(define (>> n i) (if (zero? i) n (>> (floor->exact (/ n 2)) (- i 1))))
(define (<< n i) (if (<= i 0) (>> n (- i)) (<< (+ n n) (- i 1))))

A126309 A014486-index for the Dyck path "compressed" from the n-th Dyck path encoded by A014486(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 3, 1, 2, 1, 1, 3, 3, 3, 3, 8, 0, 1, 1, 1, 3, 1, 2, 1, 1, 3, 3, 3, 3, 8, 1, 2, 2, 2, 5, 1, 2, 1, 1, 3, 3, 3, 3, 8, 3, 6, 3, 3, 7, 3, 3, 3, 8, 8, 8, 8, 8, 22, 0, 1, 1, 1, 3, 1, 2, 1, 1, 3, 3, 3, 3, 8, 1, 2, 2, 2, 5, 1, 2, 1, 1, 3, 3, 3, 3, 8, 3, 6, 3, 3, 7, 3, 3, 3, 8
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

According to Vaillé, the concept of "compression d'un pont" was introduced by Poupard, in "Sur les quasi-ponts" paper. In effect, the operation removes all the peaks /\ from the Dyck path.

Examples

			A014486(4) encodes the Dyck path /\/\/\, of which, when all the peaks are removed, nothing remains, thus a(4)=0. A014486(18) encodes the Dyck path:
....../\
.../\/..\
../......\,
which, after the peaks are removed, results
.../\,
../..\ encoded by A014486(3), thus a(18)=3.

Links

Crossrefs

Cf. A014486, A080300, A125985, A125986, A126308, A126310.

Formula

a(n) = A080300(A126308(A014486(n))).
a(n) = A125985(A126310(A125986(n))).

A126310 A014486-index for the Dyck path "derived" from the n-th Dyck path encoded by A014486(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 1, 0, 4, 2, 2, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 0, 9, 4, 4, 4, 2, 4, 2, 4, 5, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 6, 7, 3, 2, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 1, 1, 1, 1, 0, 23, 9, 9, 9, 4, 9, 4, 9, 10, 4, 4, 4, 4, 2, 9, 4, 4, 4, 2, 9, 4, 11, 12, 5, 4, 4, 5, 2, 4, 2, 4, 5, 2, 4, 5, 5
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

According to Vaillé, the concept of "dérivation des ponts" is defined by Kreweras, in "Sur les éventails de segments" paper.

Links

Crossrefs

Cf. A125986, A126309, A125985.

Formula

a(n) = A125986(A126309(A125985(n))).
Showing 1-9 of 9 results.

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