A055091 -id:A055091 - cp's OEIS frontend

A034968 Minimal number of factorials that add to n.

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Erich Friedman

nonn

Equivalently, sum of digits when n is written in factorial base (A007623).
Equivalently, a(0)...a(n!-1) give the total number of inversions of the permutations of n elements in lexicographic order (the factorial numbers in rising base are the inversion tables of the permutations and their sum of digits give the total number of inversions, see example and the Fxtbook link). - Joerg Arndt, Jun 17 2011
Also minimum number of adjacent transpositions needed to produce each permutation in the list A055089, or number of swappings needed to bubble sort each such permutation. (See A055091 for the minimum number of any transpositions.)

			a(205) = a(1!*1 + 3!*2 + 4!*3 + 5!*1) = 1+2+3+1 = 7. [corrected by Shin-Fu Tsai, Mar 23 2021]
From _Joerg Arndt_, Jun 17 2011: (Start)
   n:    permutation   inv. table a(n)  cycles
   0:    [ 0 1 2 3 ]   [ 0 0 0 ]   0    (0) (1) (2) (3)
   1:    [ 0 1 3 2 ]   [ 0 0 1 ]   1    (0) (1) (2, 3)
   2:    [ 0 2 1 3 ]   [ 0 1 0 ]   1    (0) (1, 2) (3)
   3:    [ 0 2 3 1 ]   [ 0 1 1 ]   2    (0) (1, 2, 3)
   4:    [ 0 3 1 2 ]   [ 0 2 0 ]   2    (0) (1, 3, 2)
   5:    [ 0 3 2 1 ]   [ 0 2 1 ]   3    (0) (1, 3) (2)
   6:    [ 1 0 2 3 ]   [ 1 0 0 ]   1    (0, 1) (2) (3)
   7:    [ 1 0 3 2 ]   [ 1 0 1 ]   2    (0, 1) (2, 3)
   8:    [ 1 2 0 3 ]   [ 1 1 0 ]   2    (0, 1, 2) (3)
   9:    [ 1 2 3 0 ]   [ 1 1 1 ]   3    (0, 1, 2, 3)
  10:    [ 1 3 0 2 ]   [ 1 2 0 ]   3    (0, 1, 3, 2)
  11:    [ 1 3 2 0 ]   [ 1 2 1 ]   4    (0, 1, 3) (2)
  12:    [ 2 0 1 3 ]   [ 2 0 0 ]   2    (0, 2, 1) (3)
  13:    [ 2 0 3 1 ]   [ 2 0 1 ]   3    (0, 2, 3, 1)
  14:    [ 2 1 0 3 ]   [ 2 1 0 ]   3    (0, 2) (1) (3)
  15:    [ 2 1 3 0 ]   [ 2 1 1 ]   4    (0, 2, 3) (1)
  16:    [ 2 3 0 1 ]   [ 2 2 0 ]   4    (0, 2) (1, 3)
  17:    [ 2 3 1 0 ]   [ 2 2 1 ]   5    (0, 2, 1, 3)
  18:    [ 3 0 1 2 ]   [ 3 0 0 ]   3    (0, 3, 2, 1)
  19:    [ 3 0 2 1 ]   [ 3 0 1 ]   4    (0, 3, 1) (2)
  20:    [ 3 1 0 2 ]   [ 3 1 0 ]   4    (0, 3, 2) (1)
  21:    [ 3 1 2 0 ]   [ 3 1 1 ]   5    (0, 3) (1) (2)
  22:    [ 3 2 0 1 ]   [ 3 2 0 ]   5    (0, 3, 1, 2)
  23:    [ 3 2 1 0 ]   [ 3 2 1 ]   6    (0, 3) (1, 2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Joerg Arndt, Matters Computational (The Fxtbook), fig.10-1.B on p.234.
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
Index entries for sequences related to factorial base representation

Cf. A368342 (partial sums), A001809 (sums of n! terms).
Cf. A000142, A007489, A007623, A033312, A055091, A139365.
Cf. A000217, A001222, A056019, A060130, A099563, A225901, A257684, A257687, A257694, A276076, A276146.
Cf. A227148 (positions of even terms), A227149 (of odd terms).
Cf. also A219650, A219651, A219666, A230423.
Differs from analogous A276150 for the first time at n=24.
Positions of records are A200748.

[seq(convert(fac_base(j),`+`),j=0..119)]; # fac_base and PermRevLexUnrank given in A055089. Perm2InversionVector in A064039
Or alternatively: [seq(convert(Perm2InversionVector(PermRevLexUnrank(j)),`+`),j=0..119)];
# third Maple program:
b:= proc(n, i) local q;
      `if`(n=0, 0, b(irem(n, i!, 'q'), i-1)+q)
    end:
a:= proc(n) local k;
      for k while k!Alois P. Heinz, Nov 15 2012

a[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 06 2013, after Benoit Cloitre *)

a(n)=local(k,r);k=2;r=0;while(n>0,r+=n%k;n\=k;k++);r \\ Franklin T. Adams-Watters, May 13 2009

def a(n):
    k=2
    r=0
    while n>0:
        r+=n%k
        n=n//k
        k+=1
    return r
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 19 2017, after PARI program

def A034968(n, p=2): return n if nA034968(n//p, p+1) + n%p
print([A034968(n) for n in range(106)]) # Michael S. Branicky, Oct 27 2024

(define (A034968 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (remainder n i)))))))
;; Antti Karttunen, Aug 29 2016

a(n) = n - Sum_{i>=2} (i-1)*floor(n/i!). - Benoit Cloitre, Aug 26 2003
G.f.: 1/(1-x)*Sum_{k>0} (Sum_{i=1..k} i*x^(i*k!))/(Sum_{i=0..k} x^(i*k!)). - Franklin T. Adams-Watters, May 13 2009
From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A099563(n) + a(A257687(n)).
a(0) = 0; for n >= 1, a(n) = A060130(n) + a(A257684(n)).
Other identities. For all n >= 0:
a(n) = A001222(A276076(n)).
a(n) = A276146(A225901(n)).
a(A000142(n)) = 1, a(A007489(n)) = n, a(A033312(n+1)) = A000217(n).
a(A056019(n)) = a(n).
A219651(n) = n - a(n).
(End)

Additional comments from Antti Karttunen, Aug 23 2001

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

A060130 Number of nonzero digits in factorial base representation (A007623) of n; minimum number of transpositions needed to compose each permutation in the lists A060117 & A060118.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 02 2001

nonn

			19 = 3*(3!) + 0*(2!) + 1*(1!), thus it is written as "301" in factorial base (A007623). The count of nonzero digits in that representation is 2, so a(19) = 2.

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

Cf. A007623, A034968, A055091, A060117, A060118, A060128, A060129, A060131, A060502, A257687, A275734, A275735, A276076.
Cf. A227130 (positions of even terms), A227132 (of odd terms).
Cf. also A225901, A232094, A257694, A257695.
The topmost row and the leftmost column in array A230415, the left edge of triangle A230417.
Differs from similar A267263 for the first time at n=30.

A060130(n) = count_nonfixed(convert(PermUnrank3R(n), 'disjcyc'))-nops(convert(PermUnrank3R(n), 'disjcyc')) or nops(fac_base(n))-nops(positions(0, fac_base(n)))
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;
count_nonfixed := l -> convert(map(nops, l), `+`);
positions := proc(e, ll) local a, k, l, m; l := ll; m := 1; a := []; while(member(e, l[m..nops(l)], 'k')) do a := [op(a), (k+m-1)]; m := k+m; od; RETURN(a); end;
# For procedure PermUnrank3R see A060117

Block[{nn = 105, r}, r = MixedRadix[Reverse@ Range[2, -1 + SelectFirst[Range@ 12, #! > nn &]]]; Array[Count[IntegerDigits[#, r], k_ /; k > 0] &, nn, 0]] (* Michael De Vlieger, Dec 30 2017 *)

(define (A060130 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (if (zero? (remainder n i)) 0 1)))))))
;; Two other implementations, that use memoization-macro definec:
(definec (A060130 n) (if (zero? n) n (+ 1 (A060130 (A257687 n)))))
(definec (A060130 n) (if (zero? n) n (+ (A257511 n) (A060130 (A257684 n)))))
;; Antti Karttunen, Dec 30 2017

a(0) = 0; for n > 0, a(n) = 1 + a(A257687(n)).
a(0) = 0; for n > 0, a(n) = A257511(n) + a(A257684(n)).
a(n) = A060129(n) - A060128(n).
a(n) = A084558(n) - A257510(n).
a(n) = A275946(n) + A275962(n).
a(n) = A275948(n) + A275964(n).
a(n) = A055091(A060119(n)).
a(n) = A069010(A277012(n)) = A000120(A275727(n)).
a(n) = A001221(A275733(n)) = A001222(A275733(n)).
a(n) = A001222(A275734(n)) = A001222(A275735(n)) = A001221(A276076(n)).
a(n) = A046660(A275725(n)).
a(A225901(n)) = a(n).
A257511(n) <= a(n) <= A034968(n).
A275806(n) <= a(n).
a(A275804(n)) = A060502(A275804(n)). [A275804 gives all the positions where this coincides with A060502.]
a(A276091(n)) = A260736(A276091(n)). [A276091 gives all the positions where this coincides with A260736.]

Example-section added, name edited, the old Maple-code moved away from the formula-section, and replaced with all the new formulas by Antti Karttunen, Dec 30 2017

A055093 Number of moved (non-fixed) elements in each permutation given in reversed colexicographic ordering A055089, i.e., the sum of their cycle lengths (excluding the 1-cycles, i.e., fixed elements).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 04 2000

nonn

Also number of displacements for permutations in lexicographic order. - Joerg Arndt, Jan 22 2024

Antti Karttunen, Table of n, a(n) for n = 0..40320
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)
Index entries for sequences related to factorial base representation

Cf. A055089, A055090, A055091, A055092, A060129, A275812, A290095.

A055093(n) = count_nonfixed(convert(PermRevLexUnrank(j), 'disjcyc')).
count_nonfixed := l -> convert(map(nops,l), `+`);
# Procedure PermRevLexUnrank given in A055089.

a(n) = A055090(n) + A055091(n).

a(n) = A275812(A290095(n)) = A060129(A060126(n)). - Antti Karttunen, Dec 30 2017

Entry revised by Antti Karttunen, Dec 30 2017

A227148 Numbers k for which the sum of digits is even when k is written in the factorial base (A007623).

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 25, 26, 29, 30, 33, 34, 37, 38, 41, 42, 45, 46, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 73, 74, 77, 78, 81, 82, 85, 86, 89, 90, 93, 94, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 121, 122, 125

Offset: 1

Antti Karttunen, Jul 02 2013

Numbers k for which minimal number of factorials needed to add to get k is even.
This sequence offers one possible analog to A001969 (evil numbers) in factorial base system. A227130 gives another kind of analog.
In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227149.
The sequence gives the positions of even permutations in the tables A055089 and A195663; and equivalently, the positions of even numbers in A055091.

Antti Karttunen, Table of n, a(n) for n = 1..2520
Wikipedia, Parity of permutation.

Complement: A227149. Cf. also A001969, A034968, A227130.

q[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; EvenQ[s]]; Select[Range[0, 125], q] (* Amiram Eldar, Jan 24 2024 *)

A290095 a(n) = A275725(A060126(n)); prime factorization encodings of cycle-polynomials computed for finite permutations listed in reversed colexicographic ordering.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 17 2017

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 table A055089 (A195663). See the examples.

			Consider the first eight permutations (indices 0-7) listed in A055089:
  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]
  2,3,1 [One 3-cycle, thus a(4) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(5) = 2^2 * 3^1 = 12]
  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].

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation

Cf. A055090, A055091, A055092, A055093, A060126, A290096, A290097.

Cf. also A275725, A275734, A275735, A276076 and tables A055089, A195663.

a(n) = A275725(A060126(n)).
Other identities:
A046523(a(n)) = A290096(n).
A056170(a(n)) = A055090(n).
A046660(a(n)) = A055091(n).
A072411(a(n)) = A055092(n).
A275812(a(n)) = A055093(n).

A227149 Numbers k for which the sum of digits is odd when k is written in the factorial base (A007623).

Original entry on oeis.org

1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66, 69, 70, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 97, 98, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 120, 123, 124

Offset: 1

Antti Karttunen, Jul 02 2013

Numbers k for which minimal number of factorials needed to add to get k is odd.
This sequence offers one possible analog to A000069 (odious numbers) in factorial base system. A227132 gives another kind of analog.
In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227148.
The sequence gives the positions of odd permutations in the tables A055089 and A195663; and equivalently, the positions of odd numbers in A055091.

Antti Karttunen, Table of n, a(n) for n = 1..2520
Wikipedia, Parity of permutation.
Index entries for sequences related to factorial base representation

Complement: A227148. Cf. also A000069, A034968, A055091, A227132.

Characteristic function: A374468, see also A262725.

q[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; OddQ[s]]; Select[Range[125], q] (* Amiram Eldar, Jan 24 2024 *)

A055090 Number of cycles (excluding fixed points) of the n-th finite permutation in reversed colexicographic ordering (A055089).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 18 2000

nonn

Among the first n! entries k appears A136394(n,k) times. - Tilman Piesk, Apr 06 2012

Tilman Piesk, Table of n, a(n) for n = 0..5039
Tilman Piesk, Table for A198380 (a[n] is the number of addends in parentheses on the right of this table)
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)
Index entries for sequences related to factorial base representation

Cf. A195663, A195664, A055089 (ordered finite permutations).
Cf. A198380 (cycle type of the n-th finite permutation).
Cf. A136394, A055089, A055090, A055093, A055091, A060128, A290095.

with(group); seq(nops(convert(PermRevLexUnrank(j),'disjcyc')),j=0..)];
# Procedure PermRevLexUnrank given in A055089.

a(n) = A055093(n) - A055091(n).

a(n) = A056170(A290095(n)) = A060128(A060126(n)). - Antti Karttunen, Dec 30 2017

Name changed by Tilman Piesk, Apr 06 2012

cp's OEIS Frontend

A034968 Minimal number of factorials that add to n.

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A055089 List of all finite permutations in reversed colexicographic ordering.

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A060130 Number of nonzero digits in factorial base representation (A007623) of n; minimum number of transpositions needed to compose each permutation in the lists A060117 & A060118.

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A055093 Number of moved (non-fixed) elements in each permutation given in reversed colexicographic ordering A055089, i.e., the sum of their cycle lengths (excluding the 1-cycles, i.e., fixed elements).

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A227148 Numbers k for which the sum of digits is even when k is written in the factorial base (A007623).

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A290095 a(n) = A275725(A060126(n)); prime factorization encodings of cycle-polynomials computed for finite permutations listed in reversed colexicographic ordering.

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A227149 Numbers k for which the sum of digits is odd when k is written in the factorial base (A007623).

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