0,2

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

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

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.

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 *)

(MIT/GNU Scheme, with Antti Karttunen's intseq-library):

;; Note that in Scheme, vector indexing is zero-based.

(definec (A055089 n) (vector-ref (A055089permvec-short (A220658 n)) (A220659 n)))

(definec (A055089permvec-short rank) (A055089permvec  (+ 1 (A084558 rank)) rank))

(define (A055089permvec size rank) (let ((permvec (make-initialized-vector size 1+))) (let outloop ((rank rank) (i 2)) (cond ((zero? rank) (permvec1inverse-of permvec)) (else (let inloop ((k (- i 1))) (cond ((< k (- i (remainder rank i))) (outloop (floor->exact (/ rank i)) (+ 1 i))) (else (begin (let ((tmp (vector-ref permvec (- k 1)))) (vector-set! permvec (- k 1) (vector-ref permvec k)) (vector-set! permvec k tmp)) (inloop (- k 1)))))))))))

(define (permvec1inverse-of permvec) (make-initialized-vector (vector-length permvec) (lambda (i) (permvec1find-pos-of-i-from (+ 1 i) permvec))))

(define (permvec1find-pos-of-i-from i permvec) (let loop ((k 0)) (cond ((= k (vector-length permvec)) #f) ((= i (vector-ref permvec k)) (+ 1 k)) (else (loop (+ k 1)))))) ;; Antti Karttunen, Dec 18 2012

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.

Sequence in context: A049456 A117506 A179205 * A060117 A196526 A234504

Adjacent sequences:  A055086 A055087 A055088 * A055090 A055091 A055092

nonn,tabf

Antti Karttunen, Apr 18 2000

Name changed by Tilman Piesk, Feb 01 2012

approved