A060117 -id:A060117 - cp's OEIS frontend

A275834 Positions of even terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an even cycle.

1, 4, 7, 9, 11, 15, 16, 17, 18, 20, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 51, 53, 55, 58, 66, 68, 70, 75, 76, 77, 85, 88, 90, 91, 92, 96, 98, 99, 101, 102, 104, 106, 108, 109, 110, 115, 118, 121, 124, 127, 129, 131, 135, 136, 137, 138, 140, 141, 143, 145, 148, 151, 153, 155, 159, 160, 161, 162, 164, 165, 167, 169, 172, 175, 177, 179

Offset: 1

Antti Karttunen, Aug 11 2016

nonn

Equally: positions of even terms in A275726.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Complement: A275833.
A subsequence of A273670 and A275814.
Cf. A060117, A060118, A275726, A275832.

A065183 Permutation of nonnegative integers produced when the finite permutations listed by A060117 are subjected to the inverse of (variant of) Foata's transformation. Inverse of A065184.

Original entry on oeis.org

0, 1, 2, 4, 5, 3, 6, 7, 12, 20, 19, 17, 14, 21, 8, 10, 15, 18, 23, 16, 22, 13, 11, 9, 24, 25, 26, 28, 29, 27, 48, 49, 78, 108, 103, 91, 74, 111, 62, 69, 75, 104, 101, 94, 100, 83, 71, 64, 54, 55, 80, 109, 107, 90, 30, 31, 36, 44, 43, 41, 56, 58, 72, 110, 106, 77, 59, 57, 81

Offset: 0

Antti Karttunen, Oct 19 2001

nonn

Index entries for sequences that are permutations of the natural numbers

A065161-A065163 give cycle counts and max lengths. Cf. also A065182, A065181 for Maple procedure FoataInv and A060117 for PermUnrank3R and A060125 for PermRank3R.

[seq(PermRank3R(FoataInv(PermUnrank3R(j))),j=0..119)];

A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 10 2016

			For n=27, which in factorial base (A007623) is "1011" and encodes (in A060118-order) permutation "23154" with one 3-cycle and one 2-cycle, the longest cycle has three elements, thus a(27) = 3.

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

Cf. A007623, A060117, A060118.
Cf. A051903, A275725.
Cf. A261220 (gives the positions of 1 and 2's).
Differs from A060131 for the first time at n=27, where a(27) = 3, while A060131(27) = 6.

(define (A275803 n) (A051903 (A275725 n)))

a(n) = A051903(A275725(n)).

A275813 Positions of odd terms in A060131; indices of permutations of an odd order in tables A060117 & A060118.

Original entry on oeis.org

0, 3, 5, 8, 10, 13, 14, 19, 22, 30, 33, 35, 36, 39, 41, 42, 45, 47, 50, 52, 54, 57, 59, 63, 65, 69, 71, 73, 74, 81, 83, 84, 87, 89, 93, 95, 97, 100, 105, 107, 111, 113, 114, 117, 119, 144, 147, 149, 152, 154, 157, 158, 163, 166, 168, 171, 173, 176, 178, 181, 182, 187, 190, 192, 195, 197, 200, 202, 205, 206, 211, 214, 216, 219, 221

Offset: 0

Antti Karttunen, Aug 10 2016

nonn

Indexing starts with zero, because a(0) = 0 (indicating an identity permutation) is a special case in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to factorial base representation

Complement: A275814.
Cf. A060117, A060118, A060131.
Cf. A275809 (a subsequence).

A275814 Positions of even terms in A060131; indices of permutations of an even order in tables A060117 & A060118.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 11, 12, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 37, 38, 40, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 75, 76, 77, 78, 79, 80, 82, 85, 86, 88, 90, 91, 92, 94, 96, 98, 99, 101, 102, 103, 104, 106, 108, 109, 110, 112, 115, 116, 118, 120, 121, 122, 123, 124, 125

Offset: 1

Antti Karttunen, Aug 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation

Complement: A275813.
Cf. A060117, A060118, A060131.
Cf. A275834 (a subsequence).

A275833 Positions of odd terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an odd cycle.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 10, 12, 13, 14, 19, 22, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 52, 54, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 69, 71, 72, 73, 74, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 93, 94, 95, 97, 100, 103, 105, 107, 111, 112, 113, 114, 116, 117, 119, 120, 122, 123, 125, 126, 128, 130, 132, 133, 134, 139, 142, 144

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

Indexing starts from zero, because a(0)=0 is a special case in this sequence.

Equally: positions of odd terms in A275726.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Complement: A275834.
Cf. A153880 (a subsequence).
Cf. A275726, A275832.

A064039 Reversed inversion vectors for the permutations of A060117, presented as pseudo-decimal numbers.

Original entry on oeis.org

0, 1, 10, 11, 21, 20, 100, 101, 110, 111, 121, 120, 210, 211, 200, 201, 220, 221, 311, 310, 321, 320, 301, 300, 1000, 1001, 1010, 1011, 1021, 1020, 1100, 1101, 1110, 1111, 1121, 1120, 1210, 1211, 1200, 1201, 1220, 1221, 1311, 1310, 1321, 1320, 1301, 1300

Offset: 1

Antti Karttunen, Aug 23 2001

If one uses the ordering of A055089 instead of A060117 (procedure PermRevLexUnrank instead of PermUnrank3R) one gets A007623 (Integers written in factorial base) which is a permutation of this sequence.

SiteSwap2ToDec procedure given in A060496 and PermUnrank3R in A060117.

[seq(SiteSwap2ToDec(Perm2InversionVector(PermUnrank3R(j))),j=0..119)];
Perm2InversionVector := proc(p) local n,i,j,a,c; n := nops(p); a := []; for i from 2 to n do c := 0; for j from 1 to i-1 do if(p[j] > p[i] then c := c+1; fi; od; a := [op(a),c]; od; RETURN(a); end;

A055089 List of all finite permutations in reversed colexicographic ordering.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 18 2000

			In this table, each row consists of A001563(n) permutations of n+1 terms; i.e., we have (1/) 2,1/ 1,3,2; 3,1,2; 2,3,1; 3,2,1/ 1,2,4,3; 2,1,4,3; ... .
Append to each an infinite number of fixed terms and we get a list of rearrangements of the natural numbers, but with only a finite number of terms permuted:
1/2,3,4,5,6,7,8,9,...
2,1/3,4,5,6,7,8,9,...
1,3,2/4,5,6,7,8,9,...
3,1,2/4,5,6,7,8,9,...
2,3,1/4,5,6,7,8,9,...
3,2,1/4,5,6,7,8,9,...
1,2,4,3/5,6,7,8,9,...
2,1,4,3/5,6,7,8,9,...
Alternatively, if we take only the first n terms of each such infinite row, then the first n! rows give all permutations of the elements 1,2,...,n.

Antti Karttunen, Rows 0..719 of the irregular table, flattened.
Daniel Forgues, Tilman Piesk, et al., Orderings, OEIS Wiki.
Antti Karttunen, Ranking and unranking functions, OEIS Wiki.
Tilman Piesk, Permutations and partitions in the OEIS, Wikiversity.
Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.
Index entries for sequences related to factorial base representation
Index entries for sequences related to permutations

Inversion vectors: A007623, cycle counts: A055090, minimum number of transpositions: A055091, minimum number of adjacent transpositions: A034968, order of each permutation: A055092, number of non-fixed elements: A055093, positions of inverses: A056019, positions after Foata transform: A065181; positions of fixed-point-free involutions: A064640.
Cf. A057112, A060112, A060117, A060118, A060132, A060133.
Cf. A195663, array of the infinite rows.
This permutation list gives essentially the same information as A030298/A030299, but in a more compact way, by skipping those permutations of A030298 that start with a fixed element.
A220658(n) gives the rank r of the permutation of which the term at a(n) is an element.
A220659(n) gives the zero-based position (from the left) of that a(n) in that permutation of rank r.
A084558(r)+1 gives the size of the finite subsequence (of the r-th infinite, but finitary permutation) which has been included in this list.

factorial_base := proc(nn) local n,a,d,j,f; n := nn; if(0 = n) then RETURN([0]); fi; a := []; f := 1; j := 2; while(n > 0) do d := floor(`mod`(n,(j*f))/f); a := [d,op(a)]; n := n - (d*f); f := j*f; j := j+1; od; RETURN(a); end;
fexlist2permlist := proc(a) local n,b,j; n := nops(a); if(0 = n) then RETURN([1]); fi; b := fexlist2permlist(cdr(a)); for j from 1 to n do if(b[j] >= ((n+1)-a[1])) then b[j] := b[j]+1; fi; od; RETURN([op(b),(n+1)-a[1]]); end;
fac_base := n -> fac_base_aux(n,2); fac_base_aux := proc(n,i) if(0 = n) then RETURN([]); else RETURN([op(fac_base_aux(floor(n/i),i+1)), (n mod i)]); fi; end;
PermRevLexUnrank := n -> `if`((0 = n),[1],fexlist2permlist(fac_base(n)));
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]); fi; end; # "the tail of the list"
# Same algorithm in different guise, showing how permutations are composed of adjacent transpositions (compare to algorithm PermUnrank3R at A060117):
PermRevLexUnrankAMSDaux := proc(n,r, pp) local s,p,k; p := pp; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); for k from n-s to n-1 do p := permul(p,[[k,k+1]]); od; RETURN(PermRevLexUnrankAMSDaux(n-1, r-(s*((n-1)!)), p)); fi; end;
PermRevLexUnrankAMSD := proc(r) local n; n := nops(factorial_base(r)); convert(PermRevLexUnrankAMSDaux(n+1,r,[]),'permlist',1+(((r+2) mod (r+1))*n)); end;

A055089L[n_] := Reverse@SortBy[DeleteCases[Permutations@Range@n, {, n}], Reverse]; Flatten@Array[A055089L, 4] (* JungHwan Min, Aug 28 2016 *)

[seq(op(PermRevLexUnrank(j)), j=0..)]; (see Maple code given below).

Name changed by Tilman Piesk, Feb 01 2012

A255411 Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

Original entry on oeis.org

0, 4, 12, 16, 18, 22, 48, 52, 60, 64, 66, 70, 72, 76, 84, 88, 90, 94, 96, 100, 108, 112, 114, 118, 240, 244, 252, 256, 258, 262, 288, 292, 300, 304, 306, 310, 312, 316, 324, 328, 330, 334, 336, 340, 348, 352, 354, 358, 360, 364, 372, 376, 378, 382, 408, 412, 420, 424, 426, 430, 432, 436, 444

Offset: 0

Antti Karttunen, Apr 16 2015

Nonnegative integers such that the number of ones (A257511) in their factorial base representation (A007623) is zero.
Nonnegative integers such that the least missing nonzero digit (A257079) in their factorial base representation is one.
a(n) can be also directly computed from n by "shifting left" its factorial base representation (that is, by appending one zero to the right, see A153880) and then incrementing all nonzero digits by one, and then converting the resulting (still valid) factorial base number back to decimal. See the examples.
The sequences A227130 and A227132 are closed under a(n), in other words, permutation listed as the a(n)-th entry in tables A060117 & A060118 has the same parity as the n-th entry in those same tables.

			Factorial base representation (A007623) of 1 is "1", shifting it left yields "10", and when we increment all nonzero digits by one, we get "20", which is the factorial base representation of 4 (as 4 = 2*2! + 0*1!), thus a(1) = 4.
F.b.r. of 2 is "10", shifting it left yields "100", and "200" is f.b.r. of 12, thus a(2) = 12.
F.b.r. of 43 is "1301", shifting it left and incrementing all nonzeros by one yields "24020", which is f.b.r of 340, thus a(43) = 340.

Antti Karttunen, Table of n, a(n) for n = 0..40319

Complement: A256450.
Positions of ones in A257079, fixed points of A257080, positions of zeros in A257511, A257081 and A257261.
Cf. A007623, A227157, A153880, A231720.
Cf. A255341, A255342, A255343.
Cf. A257262, A257263.
Cf. also A227130/A227132, A060117/A060118 and also arrays A257503 & A257505.

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 500}]; {0}~Join~Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 0](* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
Select[Range[0, 444], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@Range[2, 12]]; Table[FromDigits[Map[If[# == 0, 0, # + 1] &, IntegerDigits[n, r]]~Join~{0}, r], {n, 0, 60}] (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

A030299 Decimal representation of permutations of lengths 1, 2, 3, ... arranged lexicographically.

Original entry on oeis.org

1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321, 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425

Offset: 1

N. J. A. Sloane and Clark Kimberling

This is a list of the permutations in "one-line" notation (cf. Dixon and Mortimer, p. 2). The i-th element of the string is the image of i under the permutation. For example 231 is the permutation that sends 1 to 2, 2 to 3, and 3 to 1. - N. J. A. Sloane, Apr 12 2014
Precise definition of the term "Decimal representation" (required for indices n>409113): Numbers N(s) = Sum_{i=1..m} s(i)*10^(m-i), where s runs over the permutations of (1,...,m), and m=1,2,3,.... This also defines the "lexicographical" order: Obviously 21 comes before 123, etc. The lexicographical order of the permutations, for given m, is the same as the natural order of the numbers N(s). - M. F. Hasler, Jan 28 2013
An alternate variant, using concatenation of the permutations, is very clumsy once the length exceeds 9. For example, after 987654321 (= A030299(409113), where 409113 = A007489(9)) we would get 12345678910, 12345678109, ... In A030298 this problem has been avoided by listing the elements of permutations as separate terms. [Edited by M. F. Hasler, Jan 28 2013]
Sequence A051845 is a base-independent version of this sequence: Permutations of 1...m are considered as numbers written in base m+1. - M. F. Hasler, Jan 28 2013

John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7 MR1409812 (98m:20003).

Antti Karttunen, Table of n, a(n) for n = 1..5913
OEIS Wiki, Discussion about alternate definition(s) of this sequence, started by _M. F. Hasler_, Jan 28 2013
Index entries for sequences related to permutations
Index entries for sequences which agree for a long time but are different

A007489(n) gives the position (index) of the term corresponding to last permutation of n elements: (n,n-1,...,1).
The first differences A220664 has interesting fractal structure, see A219664 and A217626.
Cf. also A030298, A055089, A060117, A181073, A352991 (by concatenation).
See A240763 for preferential arrangements.

seq(seq(add(s[i]*10^(m-i),i=1..m),s=combinat:-permute([$1..m])),m=1..5); # Robert Israel, Oct 14 2015

Flatten @ Table[FromDigits /@ Permutations[Table[i,{i,n}]],{n,9}] (* For first 409113 terms; Zak Seidov, Oct 03 2015 *)

is_A030299(n)={ (n>1234567890 & print("maybe")) || vecsort(digits(n))==vector(#Str(n),i,i) } \\ /* use digits(n)=eval(Vec(Str(n))) in older versions lacking this function */ \\ M. F. Hasler, Dec 12 2012
(MIT/GNU Scheme)
;; Antti Karttunen, Dec 18 2012
;; Requires also code from A030298 and A055089:
(define (A030299 n) (vector->base-k (A030298permvec (A084556 n) (A220660 n)) 10))
(define (vector->base-k vec k) (let loop ((i 0) (s 0)) (cond ((= (vector-length vec) i) s) ((>= (vector-ref vec i) k) (error (format #f "Cannot interpret vector ~a in base ~a!" vec k))) (else (loop (+ i 1) (+ (* k s) (vector-ref vec i)))))))

from itertools import permutations
def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
def agen():
  m = 1
  while True:
    for s in permutations(range(1, m+1)): yield pmap(s, m)
    m += 1
def aupton(terms):
  alst, g = [], agen()
  while len(alst) < terms: alst += [next(g)]
  return alst
print(aupton(42)) # Michael S. Branicky, Jan 12 2021

Edited by N. J. A. Sloane, Feb 23 2010

cp's OEIS Frontend

A275834 Positions of even terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an even cycle.

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A065183 Permutation of nonnegative integers produced when the finite permutations listed by A060117 are subjected to the inverse of (variant of) Foata's transformation. Inverse of A065184.

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A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

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A275813 Positions of odd terms in A060131; indices of permutations of an odd order in tables A060117 & A060118.

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A275814 Positions of even terms in A060131; indices of permutations of an even order in tables A060117 & A060118.

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A275833 Positions of odd terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an odd cycle.

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A064039 Reversed inversion vectors for the permutations of A060117, presented as pseudo-decimal numbers.

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Maple

A055089 List of all finite permutations in reversed colexicographic ordering.

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A255411 Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

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A030299 Decimal representation of permutations of lengths 1, 2, 3, ... arranged lexicographically.

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