A209641 -id:A209641 - cp's OEIS frontend

A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 25, 22, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 41, 49, 38, 42, 50, 44, 52, 56, 39, 43, 51, 45, 53, 57, 46, 54, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66, 68, 72, 80, 96, 67, 69, 73, 81, 97, 70, 74, 82, 98, 76, 84, 100, 88, 104, 112, 71, 75, 83

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

Conjecture: For all n, a(A054429(n)) = A054429(a(n)), i.e. A054429 acts as a homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.
From Gus Wiseman, Aug 24 2021: (Start)
As a triangle with row lengths 2^n, T(n,k) for n > 0 appears (verified up to n = 2^15) to be the unique nonnegative integer whose binary indices are the k-th subset of {1..n} containing n. Here, a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion, and sets are sorted first by length, then lexicographically. For example, the triangle begins:
   1
   2  3
   4  5  6  7
   8  9 10 12 11 13 14 15
  16 17 18 20 24 19 21 25 22 26 28 23 27 29 30 31
Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]
Row lengths are A000079 (shifted right). Also Column k = 1.
Row sums are A010036.
Using reverse-lexicographic order gives A059893.
Using lexicographic order gives A059894.
Taking binary indices to prime indices gives A339195 (or A019565).
The ordering of sets is A344084.
A version using Heinz numbers is A344085.
(End)

			From _Gus Wiseman_, Aug 24 2021: (Start)
The terms, their binary expansions, and their binary indices begin:
   0:      ~ {}
   1:    1 ~ {1}
   2:   10 ~ {2}
   3:   11 ~ {1,2}
   4:  100 ~ {3}
   5:  101 ~ {1,3}
   6:  110 ~ {2,3}
   7:  111 ~ {1,2,3}
   8: 1000 ~ {4}
   9: 1001 ~ {1,4}
  10: 1010 ~ {2,4}
  12: 1100 ~ {3,4}
  11: 1011 ~ {1,2,4}
  13: 1101 ~ {1,3,4}
  14: 1110 ~ {2,3,4}
  15: 1111 ~ {1,2,3,4}
(End)

Antti Karttunen, Table of n, a(n) for n = 0..32767
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A209861. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

Cf. A010036, A026793, A048793, A111059, A147655, A246688, A246867, A261144, A272020, A294648, A339360.

a(n) = A209859(A036044(A209641(n))) = A209859(A056539(A209641(n))).

A209861 Permutation of nonnegative integers which maps A209641 to A209642.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 21, 19, 22, 24, 27, 20, 23, 25, 28, 26, 29, 30, 31, 32, 33, 34, 38, 35, 39, 42, 48, 36, 40, 43, 49, 45, 51, 54, 58, 37, 41, 44, 50, 46, 52, 55, 59, 47, 53, 56, 60, 57, 61, 62, 63, 64, 65, 66, 71, 67, 72, 76, 86, 68, 73, 77, 87, 80, 90, 96, 106, 69, 74, 78, 88, 81, 91, 97, 107, 83

Offset: 0

Antti Karttunen, Mar 24 2012

Conjecture: For all n, A209861(A054429(n)) = A054429(A209861(n)), i.e. A054429 acts as an homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.

The scatterplot graph of the sequence has a nice texture and interesting pattern.

Antti Karttunen, Table of n, a(n) for n = 0..32767
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A209862. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

a(n) = A209640(A209642(n)).

A209640 Global ranking function for restricted totally balanced binary strings given in A209641.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

The given Scheme-program implements a ranking function for the terms of A209641, using Khayyam's triangle A007318.

			a(12)=3, as 12 occurs as the 3rd term (zero-based) in A209641.
a(14)=0, as 14 doesn't occur in A209641.

This is an inverse function for A209641 in the sense that a(A209641(n)) = n for all n. The beginning of sequence coincides with A080300, because A209641 is a subsequence of A014486. Used to compute the permutation A209861.

(define (A209640 n) (if (or (zero? n) (not (member_of_A209641? n))) 0 (let* ((w (/ (binwidth n) 2))) (let loop ((rank 0) (row 1) (u (- w 1)) (n (- n (A053644 n))) (i (/ (A053644 n) 2)) (first_0_found? #f)) (cond ((or (zero? row) (zero? u) (zero? n)) (+ (expt 2 (-1+ w)) rank)) ((> i n) (loop rank (- row 1) u n (/ i 2) #t)) (else (loop (+ rank (if first_0_found? (A007318tr (- (+ row u) 1) (- row 1)) (A007318tr (- w 1) (- row 1)))) (+ row 1) (- u 1) (- n i) (/ i 2) first_0_found?)))))))
(define (binwidth n) (let loop ((n n) (i 0)) (if (zero? n) i (loop (floor->exact (/ n 2)) (1+ i)))))

A209642 A014486-codes for rooted plane trees where non-leaf branching can occur only at the leftmost branch of any level, but nowhere else. Reflected from the corresponding rightward branching codes in A071162, thus not in ascending order.

Original entry on oeis.org

0, 2, 10, 12, 42, 50, 52, 56, 170, 202, 210, 226, 212, 228, 232, 240, 682, 810, 842, 906, 850, 914, 930, 962, 852, 916, 932, 964, 936, 968, 976, 992, 2730, 3242, 3370, 3626, 3402, 3658, 3722, 3850, 3410, 3666, 3730, 3858, 3746, 3874, 3906, 3970, 3412, 3668, 3732, 3860, 3748, 3876, 3908, 3972, 3752, 3880, 3912, 3976, 3920, 3984, 4000, 4032

Offset: 0

Antti Karttunen, Mar 11 2012

Like with A071162, a(n) can be computed directly from the binary expansion of n. (See the Scheme function given). However, the function is not monotone. A209641 gives the same terms in ascending order.

Antti Karttunen, Table of n, a(n) for n = 0..32767

a(n) = A209641(A209861(n)).

def a(n):
    s=0
    i=1
    while n!=0:
        if n%2==0:
            n//=2
            s=4*s + 1
        else:
            n=(n - 1)//2
            s=(s + i)*2
        i*=4
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, May 25 2017, translated from Antti Karttunen's SCHEME code

(define (A209642 n) (let loop ((n n) (s 0) (i 1)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ (* 4 s) 1) (* i 4))) (else (loop (/ (- n 1) 2) (* 2 (+ s i)) (* i 4))))))

a(n) = A056539(A071162(n)) = A036044(A071162(n)). (See also the given Scheme-function).

A209643 A014486-indices for rooted plane trees where non-leaf branching can occur only at the leftmost branch of any level, but nowhere else.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 17, 19, 20, 21, 22, 23, 37, 42, 44, 45, 51, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 107, 121, 126, 128, 129, 149, 154, 156, 157, 163, 165, 166, 168, 169, 170, 177, 179, 180, 182, 183, 184, 186, 187, 188, 189, 191, 192, 193, 194, 195, 196, 197, 329, 371, 385, 390, 392, 393, 461, 475, 480, 482

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

5 is not member of this sequence, as it encodes in A014486 a general tree:
..|
\/
which has a nonempty branch which is not the leftmost child of its parent vertex.

Antti Karttunen, Table of n, a(n) for n = 0..32767
A. Karttunen, Corresponding trees illustrated on 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 17th, etc. row in this illustration.

Cf. A209644, A071163.

a(n) = A080300(A209641(n)).

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A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

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A209861 Permutation of nonnegative integers which maps A209641 to A209642.

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A209640 Global ranking function for restricted totally balanced binary strings given in A209641.

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A209642 A014486-codes for rooted plane trees where non-leaf branching can occur only at the leftmost branch of any level, but nowhere else. Reflected from the corresponding rightward branching codes in A071162, thus not in ascending order.

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A209643 A014486-indices for rooted plane trees where non-leaf branching can occur only at the leftmost branch of any level, but nowhere else.

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Author

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