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 1-10 of 13 results. Next

A126313 Signature-permutation of a Catalan automorphism: composition of A069772 and A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 5, 6, 4, 7, 22, 13, 15, 12, 14, 19, 21, 16, 11, 18, 10, 20, 17, 9, 64, 36, 41, 35, 40, 52, 53, 38, 34, 39, 55, 51, 37, 54, 60, 63, 32, 62, 31, 56, 59, 47, 33, 50, 27, 58, 49, 26, 43, 44, 29, 61, 30, 24, 57, 48, 25, 46, 42, 28, 23, 45, 196, 106, 120, 105, 119
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126314. a(n) = A069772(A125976(n)) = A126290(A069772(n)) = A126315(A057164(n)). The number of cycles, number of fixed points, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127277, A127278, A127279 and A127280. The fixed points are given by A127306. Note the curiosity: this automorphism partitions the A000108(8) = 1430 Catalan structures of size eight (e.g. Dyck paths of length 16) into 79 equivalence classes, of which the largest contains 79 members.

A125977 Signature-permutation of a Catalan automorphism: composition of A057163 and A125976.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 14, 16, 11, 19, 17, 20, 12, 10, 15, 21, 18, 13, 22, 23, 37, 42, 28, 51, 44, 53, 30, 25, 39, 56, 47, 33, 60, 45, 54, 57, 48, 61, 31, 40, 26, 24, 38, 43, 34, 29, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 107, 121, 79, 149
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Links

Crossrefs

Inverse: A125978. a(n) = A057163(A125976(n)). 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 A126317, A126318 and A126319. The number of fixed points seems to be given by A123050 and fixed points themselves are probably given by A126312. Cf. also A126313-A126316.
Differs from A071661 for the first time at n=43, where a(n)=40, while A071661(43)=34. Differs from A071666 for the first time at n=34, where a(n)=47, while A071666(34)=48.

A126314 Signature-permutation of a Catalan automorphism: composition of A125976 and A069772.

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 8, 4, 22, 19, 17, 12, 10, 13, 11, 16, 21, 18, 14, 20, 15, 9, 63, 56, 59, 50, 47, 62, 53, 55, 41, 39, 45, 31, 26, 24, 35, 30, 32, 27, 25, 61, 51, 52, 64, 60, 44, 58, 49, 46, 34, 28, 29, 36, 33, 42, 57, 48, 43, 37, 54, 40, 38, 23, 196, 178, 191, 192, 177
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126313. a(n) = A125976(A069772(n)) = A069772(A126290(n)) = A057164(A126316(n)).

A126315 Signature-permutation of a Catalan automorphism: composition of A069771 and A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 5, 4, 7, 22, 19, 15, 16, 10, 13, 21, 12, 11, 20, 14, 18, 17, 9, 64, 60, 52, 56, 43, 41, 32, 38, 47, 29, 55, 27, 24, 46, 36, 63, 53, 59, 44, 35, 62, 34, 33, 61, 51, 58, 57, 42, 40, 31, 39, 50, 30, 37, 49, 48, 28, 54, 26, 25, 23, 45, 196, 191, 178, 186, 164
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126316. a(n) = A069771(A125976(n)) = A126290(A069771(n)) = A126313(A057164(n)). The number of cycles, number of fixed points and maximum cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127281, A127282 and A127283. See also the comment at A127280.

A126316 Signature-permutation of a Catalan automorphism: composition of A125976 and A069771.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 8, 4, 22, 13, 17, 16, 14, 19, 11, 12, 21, 20, 10, 18, 15, 9, 63, 35, 62, 61, 34, 59, 32, 55, 52, 29, 45, 44, 42, 37, 56, 30, 53, 51, 28, 50, 27, 41, 64, 36, 31, 58, 57, 54, 47, 25, 39, 60, 33, 26, 49, 48, 40, 24, 46, 43, 38, 23, 196, 120, 106, 148, 78
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126315. a(n) = A125976(A069771(n)) = A069771(A126290(n)) = A057164(A126314(n)).

A125978 Signature-permutation of a Catalan automorphism: composition of A125976 and A057163.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 12, 16, 21, 10, 18, 11, 14, 20, 13, 15, 19, 22, 23, 45, 31, 44, 58, 26, 49, 30, 42, 57, 35, 48, 56, 63, 24, 46, 32, 43, 59, 25, 47, 28, 37, 54, 34, 40, 53, 62, 27, 50, 29, 38, 55, 33, 39, 51, 61, 36, 41, 52, 60, 64, 65, 129, 87, 128, 170, 73
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Links

Crossrefs

Inverse: A125977. a(n) = A125976(A057163(n)).

A126290 Signature-permutation of a Catalan automorphism: A069772-conjugate of A125976.

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 5, 8, 7, 22, 13, 17, 20, 10, 19, 21, 18, 11, 16, 14, 12, 15, 9, 33, 50, 47, 56, 59, 34, 39, 38, 52, 53, 23, 28, 42, 46, 61, 30, 29, 43, 44, 35, 40, 41, 64, 36, 25, 48, 57, 24, 62, 31, 32, 60, 63, 26, 49, 58, 27, 54, 37, 51, 55, 45, 196, 120, 106, 105, 119
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

This is a self-inverse permutation (involution) of the nonnegative integers.

Links

Crossrefs

a(n) = A069772(A125976(A069772(n))) = A069771(A125976(A069771(n))).

A125974 Function whose restriction to A014486 induces signature-permutation A125976.

Original entry on oeis.org

0, 1, 2, 5, 10, 3, 2, 21, 42, 11, 12, 13, 10, 11, 10, 85, 170, 43, 44, 13, 52, 7, 6, 53, 42, 11, 12, 45, 10, 43, 42, 341, 682, 171, 172, 45, 180, 39, 38, 53, 212, 23, 56, 57, 50, 51, 22, 213, 170, 43, 44, 45, 52, 39, 38, 181, 42, 43, 44, 173, 42, 171, 170, 1365, 2730
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

A125975 gives the terms i, for which a(a(i)) = i. Question: would it be possible to construct a more elegant and natural variant which were an involution for all the natural numbers? (and acting in the same way on the set A125975, or at least on the set A014486.)

Crossrefs

Python code uses the following functions: A000265, A007814 and A036987.

Programs

  • Python
    def A125974(n):
    if 0 == n:
        return n
    chosen = A000265(n)  # Initially ones, get rid of lsb-0's.
    others = n >> A007814(n + 1)  # Initially zeros, get rid of lsb-1's.
    s = 0  # the resulting sum
    b = n % 2  # n's parity.
    p = 1  # powers of two.
    while (chosen != 0) or (others != 0):
        if (1 == chosen) or (1 == A036987(chosen + 1)):  # Last one or zero at hand.
            chosen = others
            others = 0
            nb = 1 - b
        elif (0 == (chosen % 4)) or (
            3 == (chosen % 4)
        ):  # Source run continues, dest changes.
            tmp = chosen
            chosen = others
            others = tmp >> 1
            nb = 1 - b
        elif 1 == (
            chosen % 4
        ):  # Source run changes, from ones to zeros, skip past zeros.
            chosen = A000265(chosen - 1)
            nb = b
        else:  # Source run changes, from zeros to ones, skip past ones.
            chosen = chosen >> A007814(chosen + 2)
            nb = b
        s += b * p
        p <<= 1
        b = nb
    return s
  • Scheme
    (define (A125974 n) (let ((runlens (binexp->runcount1list n))) (let loop ((chosen (reverse! (bisect runlens 0))) (others (reverse! (bisect runlens 1))) (s 0) (b (modulo n 2)) (p 1)) (cond ((and (null? chosen) (null? others)) s) ((and (pair? chosen) (= 1 (car chosen)) (pair? (cdr chosen))) (loop (cdr chosen) others (+ s (* b p)) b (+ p p))) (else (loop others (if (or (null? chosen) (= 1 (car chosen))) '() (cons (- (car chosen) 1) (cdr chosen))) (+ s (* b p)) (- 1 b) (+ p p)))))))
  • Scheme
    (define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (+ 1 count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2))))))) ;; (binexp->runcount1list 25) returns (2 2 1)
  • Scheme
    (define (bisect lista parity) (let loop ((lista lista) (i 0) (z (list))) (cond ((null? lista) (reverse! z)) ((eq? i parity) (loop (cdr lista) (modulo (1+ i) 2) (cons (car lista) z))) (else (loop (cdr lista) (modulo (1+ i) 2) z)))))

A125980 Signature-permutation of a Catalan automorphism: A057163-conjugate of A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 5, 7, 6, 4, 22, 13, 18, 15, 10, 21, 12, 20, 19, 11, 17, 16, 14, 9, 64, 36, 50, 41, 27, 59, 32, 55, 52, 29, 46, 34, 38, 24, 63, 35, 49, 43, 26, 62, 40, 61, 60, 33, 48, 47, 39, 25, 58, 31, 57, 56, 30, 54, 53, 51, 28, 45, 44, 42, 37, 23, 196, 106, 148, 120, 78
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

This is self-inverse permutation (involution) of the nonnegative integers.

Links

Crossrefs

a(n) = A057163(A125976(A057163(n))) = A125977(A057163(n)) = A057163(A125978(n)).

A125979 Signature-permutation of a Catalan automorphism: composition of A125976 and A057164.

Original entry on oeis.org

0, 1, 3, 2, 8, 5, 6, 7, 4, 22, 13, 15, 18, 10, 19, 11, 20, 21, 12, 14, 16, 17, 9, 64, 36, 41, 50, 27, 52, 29, 55, 59, 32, 38, 43, 46, 24, 60, 33, 39, 47, 25, 61, 34, 62, 63, 35, 40, 48, 49, 26, 51, 28, 53, 56, 30, 54, 57, 58, 31, 37, 42, 44, 45, 23, 196, 106, 120, 148, 78
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

This is self-inverse permutation (involution) of the nonnegative integers.

Links

Crossrefs

a(n) = A125976(A057164(n)) = A057164(A125976(n)). Differs from A071663 for the first time at n=43, where a(n)=34, while A071663(n)=48.
Showing 1-10 of 13 results. Next

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