A188568 -id:A188568 - cp's OEIS frontend

A208233 First inverse function (numbers of rows) for pairing function A188568.

1, 1, 2, 3, 2, 1, 1, 3, 2, 4, 5, 2, 3, 4, 1, 1, 5, 3, 4, 2, 6, 7, 2, 5, 4, 3, 6, 1, 1, 7, 3, 5, 4, 6, 2, 8, 9, 2, 7, 4, 5, 6, 3, 8, 1, 1, 9, 3, 7, 5, 6, 4, 8, 2, 10, 11, 2, 9, 4, 7, 6, 5, 8, 3, 10, 1

Offset: 1

Boris Putievskiy, Jan 10 2013

			The start of the sequence as triangle array read by rows:
  1;
  1,2;
  3,2,1;
  1,3,2,4;
  5,2,3,41;
  1,5,3,4,2,6;
  7,2,5,4,3,6,1;
  ...
Row number k contains permutation numbers form 1 to k.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A188568.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i>=j:
   result= max(i,j)*((-1)**i+1)/2-min(i,j)*((-1)**i-1)/2
else:
   result=-max(i,j)*((-1)**j-1)/2+min(i,j)*((-1)**j+1)/2

a(n) = max(i,j)*((-1)^i+1)/2-min(i,j)*((-1)^i-1)/2, if i>=j

a(n) = -max(i,j)*((-1)^j-1)/2+min(i,j)*((-1)^j+1)/2, if i

where

t = floor((-1+sqrt(8*n-7))/2),

i = n-t*(t+1)/2,

j = (t*t+3*t+4)/2-n.

A208234 Second inverse function (numbers of columns) for pairing function A188568.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 4, 2, 3, 1, 1, 4, 3, 2, 5, 6, 2, 4, 3, 5, 1, 1, 6, 3, 4, 5, 2, 7, 8, 2, 6, 4, 5, 3, 7, 1, 1, 8, 3, 6, 5, 4, 7, 2, 9, 10, 2, 8, 4, 6, 5, 7, 3, 9, 1, 1, 10, 3, 8, 5, 6, 7, 4, 9, 2, 11

Offset: 1

Boris Putievskiy, Jan 10 2013

nonn

			The start of the sequence as triangle array read by rows:
  1;
  2,1;
  1,2,3;
  4,2,3,1;
  1,4,3,2,5;
  6,2,4,3,5,1;
  1,6,3,4,5,2,7;
  ...
Row number k contains permutation numbers form 1 to k.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A188568.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)//2
j=(t*t+3*t+4)//2-n
if i>=j:
    result=-max(i,j)*((-1)**i-1)/2+min(i,j)*((-1)**i+1)/2
else:
    result= max(i,j)*((-1)**j+1)/2-min(i,j)*((-1)**j-1)/2

a(n) = -max(i,j)*((-1)^i-1)/2+min(i,j)*((-1)^i+1)/2, if i>=j

a(n) = max(i,j)*((-1)^j+1)/2-min(i,j)*((-1)^j-1)/2, if i

where t = floor((-1+sqrt(8*n-7))/2), i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n.

A194982 Inverse permutation to A194981; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 9, 10, 8, 11, 13, 15, 14, 12, 16, 18, 20, 21, 19, 17, 22, 24, 26, 27, 28, 25, 23, 29, 31, 33, 34, 36, 35, 32, 30, 37, 39, 41, 42, 44, 45, 43, 40, 38, 46, 48, 50, 51, 53, 55, 54, 52, 49, 47, 56, 58, 60, 61, 63, 65, 66, 64, 62, 59, 57, 67, 69, 71

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A194980, A194981, A188568.

Mathematica
```
(* See A194981. *)
```

A194280 Inverse permutation to A081344.

Original entry on oeis.org

1, 2, 5, 3, 6, 9, 13, 8, 4, 7, 12, 18, 25, 19, 14, 10, 15, 20, 26, 33, 41, 32, 24, 17, 11, 16, 23, 31, 40, 50, 61, 51, 42, 34, 27, 21, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 29, 38, 48, 59, 71, 84, 98, 113

Offset: 1

Boris Putievskiy, Dec 23 2012

nonn

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This sequence is A188568 as table read layer by layer clockwise.
The same table A188568 read  by boustrophedon ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  2,5,3;
  6,9,13,8,4;
  7,12,18,25,19,14,10;
  15,20,26,33,41,32,24,17,11;
  16,23,31,40,50,61,51,42,34,27,21;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences that are permutations of the natural numbers

Cf. A081344, A064790, A188568.

t=int(math.sqrt(n-1))+1
i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
m=(i+j-1)*(i+j-2)/2+j

a(n) = (i+j-1)*(i+j-2)/2+j, where
i = mod(t;2)*min{t; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}
j = mod(t;2)*min{t; t^2 - n + 1} + mod(t + 1; 2)*min{t; n - (t - 1)^2},
t = int(math.sqrt(n-1))+1.

A185180 Enumeration table T(n,k) by antidiagonals. The order of the list is symmetrical movement from center to edges diagonal.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 9, 7, 8, 10, 14, 12, 11, 13, 15, 20, 18, 16, 17, 19, 21, 27, 25, 23, 22, 24, 26, 28, 35, 33, 31, 29, 30, 32, 34, 36, 44, 42, 40, 38, 37, 39, 41, 43, 45, 54, 52, 50, 48, 46, 47, 49, 51, 53, 55, 65, 63, 61, 59, 57, 56, 58, 60, 62, 64, 66, 77, 75

Offset: 1

Boris Putievskiy, Dec 26 2012

The natural numbers are grouped in chunks of 1, 2, 3, 4,... as (1), (2,3), (4,5,6), (7,8,9,10), etc and each group fills a diagonal in the table. The smallest number in a group is in A000124, the largest in A000217. Numbers in a group are placed on free spots as close as possible to the middle of the diagonal, given preference to the smaller row numbers in the table if there is a draw.
The resulting array is apparently a transposed version of A064789 (if this was also written as an array).
The order of the list table T(n,k):
if n is odd:
T (floor(n+1)/2,floor(n+1)/2), T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1)
if n is even:
T(floor(n+1)/2-1,floor(n+1)/2+1), T(floor(n+1)/2+1,floor(n+1)/2-1),...T(1,n), T(n,1).
Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

			The start of the sequence as table:
  1....2....5....9...14...20...27 ...
  3....4....7...12...18...25...33 ...
  6....8...11...16...23...31...40 ...
  10..13...17...22...29...38...48 ...
  15..19...24...30...37...46...57 ...
  21..26...32...39...47...56...67 ...
  28..34...41...49...58...68...79 ...
  ...
The start of the sequence as triangle array read by rows:
  1;
  2,   3;
  5,   4,  6;
  9,   7,  8, 10;
  14, 12, 11, 13, 15;
  20, 18, 16, 17, 19, 21;
  27, 25, 23, 22, 24, 26, 28;
. . .
Row number k (k > 1) of the triangle contains a permutation of the set of k numbers from (k^2-k+2)/2, (k^2-k+2)/2 + 1 ,...up to (k^2+k-2)/2 + 1, namely (k^2+k-2)/2, (k^2+k-2)/2 -2,...,(k^2-k+2)/2, (k^2-k+2)/2 + 2,..., (k^2+k-2)/2-1, (k^2+k-2)/2+1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A056011, A056023, A057027, A057028, A064578, A194981, A194982, A188568.

a[n_] := Module[{i, j, t}, i = n - t(t+1)/2; j = (t^2 + 3t + 4)/2 - n; t = Floor[(-1 + Sqrt[8n - 7])/2]; If[j <= i, (i(i+1) + (j-1)(j + 2i - 4))/2, (i(i+1) + (j-1)(j + 2i - 4))/2 + 2(j-i) - 1]];
Array[a, 68] (* Jean-François Alcover, Nov 21 2018, from Python *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j<=i:
   m=(i*(i+1) + (j-1)*(j+2*i-4))/2
else:
   m=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1

a(n) = (i*(i+1) + (j-1)*(j+2*i-4))/2, if j<=i, a(n)=(i*(i+1) + (j-1)*(j+2*i-4))/2 +2*(j-i)-1, if j>i, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2].

A064790 Inverse permutation to A060734.

Original entry on oeis.org

1, 3, 5, 2, 6, 9, 13, 8, 4, 10, 14, 19, 25, 18, 12, 7, 15, 20, 26, 33, 41, 32, 24, 17, 11, 21, 27, 34, 42, 51, 61, 50, 40, 31, 23, 16, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 36, 44, 53, 63, 74, 86, 99, 113, 98, 84, 71, 59, 48, 38, 29, 45, 54, 64, 75, 87, 100, 114

Offset: 1

N. J. A. Sloane, Oct 20 2001

From Boris Putievskiy, Mar 14 2013: (Start)
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and  from T(n,n) to T(n,1). This sequence is A188568 as table read  by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so. The same table A188568 read layer by layer clockwise is A194280. (End)

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  3,5,2;
  6,9,13,8,4;
  10,14,19,25,18,12,7;
  15,20,26,33,41,32,24,17,11;
  21,27,34,42,51,61,50,40,31,23,16;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's MathWorld, Pairing Function
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A064788, A188568, A194280.

a(n) = (i+j-1)*(i+j-2)/2+i, where i=min(t; t^2-n+1), j=min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1. - Boris Putievskiy, Dec 24 2012

A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 6, 3, 2, 1, 5, 5, 1, 3, 5, 4, 4, 4, 2, 2, 1, 7, 6, 8, 4, 3, 5, 7, 9, 7, 6, 5, 4, 3, 2, 1, 8, 11, 7, 11, 1, 4, 5, 7, 9, 10, 9, 5, 7, 6, 2, 4, 3, 2, 1, 15, 10, 9, 9, 14, 6, 3, 5, 7, 9, 11, 12, 8, 18, 8, 8, 7, 6, 4, 4, 3, 2, 1, 13, 12, 11, 10, 12, 17, 1, 6, 5, 7, 9, 11, 13, 14, 13, 16, 6, 10, 9, 8, 2, 6, 5, 4, 3, 2, 1

Offset: 1

Boris Putievskiy, Aug 29 2024

A208233 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

			Table begins:
    k=    1   2   3   4   5   6
  -----------------------------------
  n= 1:   1,  1,  3,  1,  5,  1, ...
  n= 2:   2,  2,  2,  3,  2,  5, ...
  n= 3:   3,  3,  1,  2,  3,  3, ...
  n= 4:   6,  5,  4,  4,  4,  4, ...
  n= 5:   5,  4,  8,  5,  1,  2, ...
  n= 6:   4,  6,  6, 11,  6,  6, ...
  n= 7:   7,  7,  7,  7, 14,  7, ...
  n= 8:   9, 11,  5,  9,  8, 17, ...
  n= 9:   8,  9,  9,  8, 12,  9, ...
  n= 10: 10, 10, 18, 10, 10, 15, ...
  n= 11: 15,  8, 11,  6, 11, 11, ...
  n= 12: 12, 12, 16, 12,  9, 13, ...
  n= 13: 13, 13, 13, 13, 13, 12, ...
  n= 14: 14, 19, 14, 23,  7, 14, ...
  n= 15: 11, 15, 15, 15, 15, 10, ...
  n= 16: 16, 17, 12, 21, 30, 16, ...
  n= 17: 20, 16, 17, 17, 17,  8, ...
  n= 18: 18, 18, 10, 19, 28, 18, ...
     ... .
In column 3, the first 3 blocks have lengths 3,6 and 9. In column 6, the first 2 blocks have lengths 6 and 12. Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  2,1;
  3,2,3;
  6,3,2,1;
  5,5,1,3,5;
  4,4,4,2,2,1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000194, A002024, A002260, A003991, A062707, A074294, A093178, A111651, A124625, A188568, A208233, A371355.

T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n*k+k^2]-k)/(2*k)]; R=n-k*(L-1)*L/2; P=(((-1)^Max[R,k*L+1-R]+1)*R-((-1)^Max[R,k*L+1-R]-1)*(k*L+1-R))/2; result=P+k*(L-1)*L/2]
Nmax=18; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

T(n,k) = P(n,k) + k*(L(n,k)-1)*L(n,k)/2 = P(n,k) + A062707(L(n-1),k), where L(n,k) = ceiling((sqrt(8*n*k+k^2)-k)/(2*k)), R(n,k) = n-k*(L(n,k)-1)*L(n,k)/2, P(n,k) = (((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))+1)*R(n,k)-((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))-1)*(k*L(n,k)+1-R(n,k)))/2.

T(n,1) = A188568(n). T(1,k) = A093178(k). T(n,n) = A124625(n). L(n,1) = A002024(n). L(n,2) = A000194(n). L(n,3) = A111651(n). L(n,4) = A371355(n). R(n,1) = A002260(n). R(n,2) = A074294(n).

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cp's OEIS Frontend

A208233 First inverse function (numbers of rows) for pairing function A188568.

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A208234 Second inverse function (numbers of columns) for pairing function A188568.

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A194982 Inverse permutation to A194981; every positive integer occurs exactly once.

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A194280 Inverse permutation to A081344.

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A185180 Enumeration table T(n,k) by antidiagonals. The order of the list is symmetrical movement from center to edges diagonal.

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A064790 Inverse permutation to A060734.

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A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

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