A275851 -id:A275851 - cp's OEIS frontend

A275852 Positions of zeros in A275851.

1, 3, 5, 7, 9, 11, 15, 16, 17, 20, 21, 23, 27, 29, 31, 33, 35, 39, 40, 41, 44, 45, 47, 55, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 79, 81, 82, 83, 87, 88, 89, 92, 93, 94, 95, 103, 104, 105, 107, 110, 111, 112, 113, 116, 117, 119, 127, 129, 131, 135, 136, 137, 140, 141, 143, 147, 149, 151, 153, 155, 159, 160, 161, 164, 165, 167, 175, 177, 179, 181

Offset: 1

Antti Karttunen, Aug 11 2016

nonn

These are indices of derangements in permutation lists A060117 & A060118 when only elements in range [1..(1+A084558(n))] are considered to be a part of the finite permutation whose rank number is n.

Antti Karttunen, Table of n, a(n) for n = 1..17008
Index entries for sequences related to factorial base representation

Cf. A060117, A060118, A084558, A275851.

Subsequence of A273670.

A275853 a(n) = A060502(n) + A275851(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 4, 3, 3, 3, 3, 2, 4, 3, 4, 4, 4, 3, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 2, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 4, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 5

Offset: 0

Antti Karttunen, Aug 15 2016

These are averages (number of balls) in siteswap-patterns constructed like in A060498, but with 0's replaced by the length of the pattern.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060498, A060500, A060501, A060502, A275849, A275851.

(define (A275853 n) (+ (A060502 n) (A275851 n)))

a(n) = A060502(n) + A275851(n).

A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

Original entry on oeis.org

2, 4, 18, 8, 12, 8, 150, 100, 54, 16, 24, 16, 90, 40, 54, 16, 36, 16, 60, 40, 36, 16, 24, 16, 1470, 980, 882, 392, 588, 392, 750, 500, 162, 32, 48, 32, 270, 80, 162, 32, 108, 32, 120, 80, 72, 32, 48, 32, 1050, 700, 378, 112, 168, 112, 750, 500, 162, 32, 48, 32, 450, 200, 162, 32, 72, 32, 300, 200, 108, 32, 48, 32, 630, 280, 378, 112, 252, 112, 450, 200

Offset: 0

Antti Karttunen, Aug 09 2016

nonn

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in tables A060117 or A060118. See the examples.

			Consider the first eight permutations (indices 0-7) listed in A060117:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]
  2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]
and also the seventeenth one at n=16 [A007623(16)=220] where we have:
  3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].

Antti Karttunen, Table of n, a(n) for n = 0..40319
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A000040, A001222, A001221, A002110, A007814, A046660, A048675, A051903, A056169, A056170, A084558, A243054, A257510, A275723, A275803, A275832.
Cf. A275807 (terms divided by 2).
Cf. also A060129, A060128, A060130, A060131, A072411, A275726, A275812, A275851.
Cf. also A275733, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

(define (A275725 n) (A275723bi (A002110 (+ 1 (A084558 n))) n)) ;; Code for A275723bi given in A275723.

a(n) = A275723(A002110(1+A084558(n)), n).
Other identities:
A001221(a(n)) = 1+A257510(n) (for all n >= 1).
A001222(a(n)) = 1+A084558(n).
A007814(a(n)) = A275832(n).
A048675(a(n)) = A275726(n).
A051903(a(n)) = A275803(n).
A056169(a(n)) = A275851(n).
A046660(a(n)) = A060130(n).
A072411(a(n)) = A060131(n).
A056170(a(n)) = A060128(n).
A275812(a(n)) = A060129(n).
a(n!) = 2 * A243054(n) = A000040(n)*A002110(n) for all n >= 1.

A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 3, 4, 4, 4, 2, 3, 4, 4, 3, 4, 2, 4, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 2, 3, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 2, 3, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 4, 4, 5, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. also A000142, A055093, A060117, A060118, A060120, A060128, A060130, A084558, A275851.

a(n) = A060128(n) + A060130(n).
From Antti Karttunen, Aug 11 2016: (Start)
a(n) = A275812(A275725(n)).
a(n) = 1 + A084558(n) - A275851(n).
Other identities. For all n >= 0:
a(n) = A055093(A060120(n)).
a(A000142(n)) = 2.
(End)

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

Original entry on oeis.org

2, 4, 12, 8, 12, 8, 60, 36, 24, 16, 24, 16, 60, 24, 24, 16, 36, 16, 60, 24, 36, 16, 24, 16, 420, 180, 180, 72, 180, 72, 120, 72, 48, 32, 48, 32, 120, 48, 48, 32, 72, 32, 120, 48, 72, 32, 48, 32, 420, 180, 120, 48, 120, 48, 120, 72, 48, 32, 48, 32, 180, 72, 48, 32, 72, 32, 180, 72, 72, 32, 48, 32, 420, 120, 120, 48, 180, 48, 180, 72, 48, 32, 72, 32, 120, 48, 48

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain sequences related to cycle-structures in finite permutations as ordered by lists A060117 / A060118 (and thus also related to factorial base representation, A007623) because it matches only with any such sequence b that can be computed as b(n) = f(A275725(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Cf. A007623, A060117, A060118, A046523, A275725.
Other filter-sequences related to factorial base: A278234, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A048764, A048765, A060129, A060130, A060131, A084558, A275803, A275851, A257510.

(define (A278225 n) (A046523 (A275725 n)))

a(n) = A046523(A275725(n)).

A275849 Number of unoccupied slopes in factorial base representation of n: a(n) = A084558(n) - A060502(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 4

Offset: 0

Antti Karttunen, Aug 15 2016

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060501, A060502, A084558.
Cf. A007489 (the indices of zeros).
Cf. also A225901, A275850, A275851, A275853.

(define (A275849 n) (- (A084558 n) (A060502 n)))

a(n) = A084558(n) - A060502(n).
Other identities. For all n >= 0:
a(n) = A275850(A225901(n)).
a(n) = A060501(n)-1. [To be proved.]

A060500 a(n) = number of drops in the n-th permutation of list A060118; the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060496(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2

Offset: 0

Antti Karttunen, Mar 22 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060125, A060129, A060501, A060502, A275849, A275851.

A060500 := avg(Perm2SiteSwap1(PermUnrank3R(n)));
# PermUnrank3R(r) gives the permutation with rank r in list A060117:
PermUnrank3R := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Raux(n+1, r, []), 'permlist', 1+(((r+2) mod (r+1))*n)); end;
PermUnrank3Raux := proc(n, r, p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Raux(n-1, r-(s*((n-1)!)), permul(p, [[n, n-s]]))); fi; end;
Perm2SiteSwap1 := proc(p) local ip, n, i, a; n := nops(p); ip := convert(invperm(convert(p, 'disjcyc')), 'permlist', n); a := []; for i from 1 to n do a := [op(a), ((ip[i]-i) mod n)]; od; RETURN(a); end;
avg := a -> (convert(a,`+`)/nops(a));

(define (A060500 n) (let ((s (+ 1 (A084558 n))) (p (A060118permvec-short n))) (let loop ((d 0) (i 1)) (if (> i s) d (loop (+ d (if (< (vector-ref p (- i 1)) i) 1 0)) (+ 1 i))))))
(define (A060118permvec-short rank) (permute-A060118 (make-initialized-vector (+ 1 (A084558 rank)) 1+) (+ 1 (A084558 rank)) rank))
(define (permute-A060118 elems size permrank) (let ((p (vector-head elems size))) (let unrankA060118 ((r permrank) (i 1)) (cond ((zero? r) p) (else (let* ((j (1+ i)) (m (modulo r j))) (cond ((not (zero? m)) (let ((org-i (vector-ref p i))) (vector-set! p i (vector-ref p (- i m))) (vector-set! p (- i m) org-i)))) (unrankA060118 (/ (- r m) j) j)))))))

From Antti Karttunen, Aug 18 2016: (Start)
The following formula reflects the original definition of computing the average, with a few unnecessary steps eliminated:
a(n) = 1/s * Sum_{i=1..s} ((i-p[i]) modulo s), where p is the permutation of rank n as ordered in the list A060117, and s is its size (the number of its elements) computed as s = 1+A084558(n).
a(n) = 1/s * Sum_{i=1..s} ((p[i]-i) modulo s). [If inverse permutations from list A060118 are used, then we just flip the order of difference that is used in the first formula].
a(n) = Sum_{i=1..s} [p[i]A060502 for the proof].
a(n) = A060502(A060125(n)).
a(n) = A060129(n) - A060502(n).
a(n) = A060501(n) - A275851(n) = 1 + A275849(n) - A275851(n).
(End)

Maple code collected together, alternative definition and new formulas added by Antti Karttunen, Aug 24 2016

cp's OEIS Frontend

A275852 Positions of zeros in A275851.

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A275853 a(n) = A060502(n) + A275851(n).

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A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

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A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

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A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

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A275849 Number of unoccupied slopes in factorial base representation of n: a(n) = A084558(n) - A060502(n).

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A060500 a(n) = number of drops in the n-th permutation of list A060118; the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060496(n).

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