A051845 -id:A051845 - cp's OEIS frontend

A358314 Triangle T(n,k) read by rows where T(2m - 1,k) = (A051845(2m - 1,k))/(2m - 1) and T(2m,k) = A051845(2m,k)/m for m > 0, k > 0.

1, 5, 7, 9, 10, 13, 15, 18, 19, 97, 99, 107, 111, 119, 121, 147, 149, 167, 173, 179, 183, 207, 211, 217, 223, 241, 243, 269, 271, 279, 283, 373, 374, 379, 381, 386, 387, 409, 410, 421, 424, 428, 430, 451, 453, 457, 460, 471

Offset: 1

Alexander R. Povolotsky, Nov 08 2022

The n-th row has n! elements.

			Triangle begins:
       k=1  k=2  k=3 ...
  n=1:   1;
  n=2:   5,   7;
  n=3:   9,  10, ...,  19;
  n=4:  97,  99, 107, ..., 283;
  n=5: 373, 374, 379, 381, ..., 471;
  ...

Cf. A051845, left edge = A221741, right edge = A221740.

T(n,1) = 4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((3-(-1)^n)*n) = A221741(n).

T(n,n!) = 4*((n-1)*(n+1)^(n+1)+1)/((3-(-1)^n)*n^3) = A221740(n).

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

A051846 Digits 1..n in strict descending order n..1 interpreted in base n+1.

Original entry on oeis.org

1, 7, 57, 586, 7465, 114381, 2054353, 42374116, 987654321, 25678050355, 736867805641, 23136292864686, 789018236134297, 29043982525261081, 1147797409030816545, 48471109094902544776, 2178347851919531492065, 103805969587115219182431

Offset: 1

Antti Karttunen, Dec 13 1999

All odd-indexed (2n+1) terms are divisible by (2n+1). See A051847.

All even-indexed (2n) terms are divisible by n. - Alexander R. Povolotsky, Oct 20 2022

			a(1) = 1,
a(2) = 2*3 + 1 = 7,
a(3) = 3*(4^2) + 2*4 + 1 = 57,
a(4) = 4*(5^3) + 3*(5^2) + 2*5 + 1 = 586.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

The right edge of A051845.

Cf. A023811, A051847, A058128, A062806, A199969.

a(n) := proc(n) local i; add(i*((n+1)^(i-1)),i=1..n); end;

Array[Sum[i*(# + 1)^(i - 1), {i, #}] &, 18] (* Michael De Vlieger, Apr 04 2024 *)

makelist(((n+1)^(n+1)*(n-1) + 1)/n^2,n,1,20); /* Martin Ettl, Jan 25 2013 */

PARI
```
a(n)=((n+1)^(n+1)*(n-1)+1)/n^2
```

def a(n): return sum((i+1)*(n+1)**i for i in range(n))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 10 2022

a(n) = Sum_{i=1..n} i*(n+1)^(i-1).
a(n) = ((n+1)^(n+1)*(n-1) + 1)/n^2 = A062806(n+1)/(n+1) - (n+1)^(n+1). - Benoit Cloitre, Sep 28 2002
a(n) = A028310(n-1) * A023811(n+1) + A199969(n+1). - M. F. Hasler, Jan 22 2013
a(n) = (n-1) * A058128(n+1) + 1. - Seiichi Manyama, Apr 10 2022

Minor edits in formulas by M. F. Hasler, Oct 11 2019

A134750 Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+1 positional system.

Original entry on oeis.org

1, 4, 5, 18, 20, 24, 28, 32, 34, 112, 115, 124, 130, 139, 142, 160, 163, 184, 193, 199, 205, 220, 226, 232, 241, 262, 265, 283, 286, 295, 301, 310, 313, 975, 979, 995, 1003, 1019, 1023, 1075, 1079, 1115, 1127, 1139, 1147, 1195, 1203, 1215, 1227, 1263, 1267

Offset: 1

Artur Jasinski, Nov 08 2007

Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+2 positional system see A051845

Cf. A134640, A051845.

a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; a (*Artur Jasinski*)

A158930 a(n) is the smallest integer not yet in the sequence with no common base-5 digit with a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 6, 10, 8, 14, 15, 7, 18, 9, 13, 20, 11, 19, 25, 17, 21, 50, 16, 22, 26, 23, 27, 24, 28, 62, 29, 63, 30, 64, 31, 52, 33, 54, 41, 60, 34, 53, 46, 65, 49, 67, 45, 68, 100, 32, 75, 36, 78, 37, 79, 56, 90, 39, 93, 35, 94, 51, 98, 55, 99, 57, 95, 61, 103, 156, 69

Offset: 1

R. J. Mathar, Mar 31 2009

Numbers of A031946 or of the 4th row of A051845 do not appear in this sequence. In base-5 notation the sequence reads 1,2,3,4,10,22,11,20,13,24,30,12,33,14,...

			The terms a(1) to a(4) are the first integers in order because they have only a single, non-common digit. a(5)=5(base10)=10(base5) does not share a digit with a(4)=4(base10)=4(base5). The numbers 6(base10)=11(base5) to 9(base10)=14(base5) are ruled out for a(6) because they share a 1 with 10(base5). The numbers 10(base10)=20(base5) and 11(base10)=21(base5) are also ruled out for a(6) because they either have a 0 or a 1 in common with a(5)=10(base5). So a(6)=12(base10)=22(base5) with no 0 or 1 is selected.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067581 (base-10), A158928 (base-3), A158929 (base-4).

for S in combinat:-powerset({$0..4}) minus {{},{$0..4}} do
  if member(0,S) then Last[S]:= 0 else Last[S]:= 1 fi od:
Next:= proc(S)
  global Last; local L, nL;
  if nops(S) = 1 then Last[S]:= Last[S]*5+S[1]; return Last[S] fi;
  Last[S]:= 1+Last[S];
  L:= convert(Last[S],base,nops(S));
  nL:= nops(L);
  if (not member(0,S)) then
   if L[-1] > 1 then
    Last[S]:= (nops(S))^nL;
    L:= [0$nL,1];
   else nL:= nL-1
   fi
  fi;
  L:= subs({seq(i-1=S[i],i=1..nops(S))},L);
  add(L[i]*5^(i-1),i=1..nL)
end proc:
Done:= {1}:
A[1]:= 1:
for n from 2 to 100 do
  S:= {$0..4} minus convert(convert(A[n-1],base,5),set);
  do
    x:= Next(S);
    if not member(x,Done) then break fi
  od;
  A[n]:= x;
  Done:= Done union {x};
od:
seq(A[i],i=1..100); # Robert Israel, Jun 25 2018

A051849 Table in which n-th row gives all compositions of n interpreted as digits in base n+1.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 21, 4, 8, 12, 16, 32, 36, 56, 156, 5, 10, 15, 20, 25, 45, 50, 55, 80, 85, 115, 260, 265, 295, 475, 1555, 6, 12, 18, 24, 30, 36, 60, 66, 72, 78, 108, 114, 120, 156, 162, 204, 402, 408, 414, 450, 456, 498, 744, 750, 792, 1086, 2802, 2808, 2850, 3144

Offset: 1

Antti Karttunen, Dec 13 1999

All terms on row n are divisible by n. See A051850.

			n-th row has length 2^(n-1) (A000079[n-1]) 1; 2, 4; 3, 6, 9, 21; 4, 8, 12, 16, 32, 36, 56, 156; 3 can be written as sum like 3, or 1+2 or 2+1 or 1+1+1. Numbers 3, 12, 21 and 111 interpreted in base 4 give the third row of table: 3,6,9,21

Cf. A051850, A051851, ...

Cf. A124734.

with(combinat); rows_upto_u := proc(u) local a,n; a := []; for n from 1 to u do a := [op(a),op(sort(map(list_in_base_b,map(op,map(permute,partition(n))),(n+1))))]; od; RETURN(a); end; # list_in_base_b given in A051845.

Definition corrected by Franklin T. Adams-Watters, Nov 20 2006

A051851 Table in which n-th row gives all partitions of n interpreted in base n+1. (A subset of A051849 with each term having a non-descending digit-sequence in base n+1).

Original entry on oeis.org

1, 2, 4, 3, 6, 21, 4, 8, 12, 32, 156, 5, 10, 15, 45, 50, 260, 1555, 6, 12, 18, 24, 60, 66, 114, 402, 408, 2802, 19608, 7, 14, 21, 28, 77, 84, 91, 147, 588, 595, 658, 4683, 4690, 37450, 299593, 8, 16, 24, 32, 40, 96, 104, 112, 184, 192, 824, 832, 840, 912, 1640

Offset: 1

Antti Karttunen, Dec 13 1999

All terms on row n are divisible by n (see A051852).

			Rows have lengths 1,2,3,5,7,11,15, etc. (A000041(n)). Terms: 1; 2, 4; 3, 6, 21; 4, 8, 12, 32, 156; 5, 10, 15, 45, 50, 260, 1555;

A subset of A051849.

with(combinat); rows_upto_u := proc(u) local a,n; a := []; for n from 1 to u do a := [op(a),op(sort(map(list_in_base_b,partition(n),(n+1))))]; od; RETURN(a); end; # list_in_base given in A051845.

cp's OEIS Frontend

A358314 Triangle T(n,k) read by rows where T(2m - 1,k) = (A051845(2m - 1,k))/(2m - 1) and T(2m,k) = A051845(2m,k)/m for m > 0, k > 0.

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

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Maple

Mathematica

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A051846 Digits 1..n in strict descending order n..1 interpreted in base n+1.

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Mathematica

Maxima

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A134750 Variant of permutational numbers with shifted digits 0->1->2->...->k+1 in k+1 positional system.

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Mathematica

A158930 a(n) is the smallest integer not yet in the sequence with no common base-5 digit with a(n-1).

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Maple

A051849 Table in which n-th row gives all compositions of n interpreted as digits in base n+1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A051851 Table in which n-th row gives all partitions of n interpreted in base n+1. (A subset of A051849 with each term having a non-descending digit-sequence in base n+1).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple