cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-19 of 19 results.

A189844 Number of ways to place n nonattacking composite pieces rook + semi-rider[3,3] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 22, 98, 534, 3334, 23724, 191820, 1704532, 16689868, 179288892, 2069311996, 25760882744, 345073745880, 4900331447624
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+3k)-p(j)<>3k for all j>=1, k>=1, j+3k<=n.
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189282, A099152, A189843, A189839, A187235

A189846 Number of ways to place n nonattacking composite pieces rook + semi-rider[4,4] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 114, 628, 4062, 30360, 251658, 2308648, 23351268, 259031232, 3091784268, 39697601392, 546982720164, 8064677125440
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+4k)-p(j)<>4k for all j>=1, k>=1, j+4k<=n
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189283, A099152, A189843, A189844, A189840, A187235

A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 9, 3, 1, 16, 48, 32, 0, 1, 25, 150, 250, 75, 15, 1, 36, 360, 1200, 1224, 288, 0, 1, 49, 735, 4165, 8869, 6321, 931, 133, 1, 64, 1344, 11648, 43136, 64512, 33024, 4096, 0, 1, 81, 2268, 27972, 160866, 423306, 469800
Offset: 1

Views

Author

Walter Trump, Mar 09 2021

Keywords

Comments

T(0,0):=1 for combinatorial reasons.
A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.
Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.

Examples

			  1;
  1,  1;
  1,  4,   0;
  1,  9,   9,   3;
  1, 16,  48,  32,  0;
  1, 25, 150, 250, 75, 15;

Links

Crossrefs

Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.

Formula

T(n,0) = 1.
T(n,1) = n^2.
T(n,2) = n^2*(n-1)*(n-2)/2.
T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.
T(2n+1,2n+1) = A006717(n).
T(2n,2n) = 0.

A189847 Number of ways to place n nonattacking composite pieces rook + semi-rider[5,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 696, 4572, 34260, 290328, 2751480, 28426056, 318900264, 3874868280, 50813711808, 716309557440, 10721493269568
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+5k)-p(j)<>5k for all j>=1, k>=1, j+5k<=n
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189284, A099152, A189843, A189844, A189846, A189841, A187235

A107683 Number of perfect Skolem sets.

Original entry on oeis.org

1, 1, 3, 11, 35, 114, 407, 1486, 5414, 19923, 74230, 278462, 1049318, 3972395, 15101658, 57607431, 220391316, 845366406, 3250192681, 12521965697
Offset: 1

Views

Author

Ralf Stephan, Jun 10 2005

Keywords

Links

Crossrefs

Cf. A004075, A060963, A099152.

A185056 Number of ways to place n nonattacking composite pieces bishop + semi-rook on an n X n board.

Original entry on oeis.org

1, 2, 5, 24, 125, 796, 5635, 48042, 453947, 4834872, 56433455, 727449366, 10099103269, 152097526360, 2449915208271, 42295879864692
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 22 2011

Keywords

Comments

Two semi-rooks do not attack each other if they are in the same column.

Links

Crossrefs

Cf. A088789, A187235, A002465, A099152, A000170.

A189848 Number of ways to place n nonattacking composite pieces rook + semi-rider[6,6] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 4920, 37488, 319644, 3033264, 31784280, 364902480, 4509620040, 59768680032, 847746532152
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+6k)-p(j)<>6k for all j>=1, k>=1, j+6k<=n
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189285, A099152, A189843, A189844, A189846, A189847, A189842, A187235

A306235 Indices in A306428 of permutations t with a finite number of nonfixed points and such that t_i - t_j <> j - i for any distinct i and j (see Comments for precise definition).

Original entry on oeis.org

0, 2, 4, 7, 8, 14, 15, 24, 28, 32, 33, 39, 48, 56, 60, 63, 64, 72, 80, 87, 96, 104, 111, 121, 122, 127, 134, 135, 138, 140, 142, 147, 150, 156, 159, 160, 168, 176, 184, 185, 192, 202, 207, 242, 246, 247, 258, 277, 296, 312, 314, 316, 318, 322, 326, 327, 333, 366, 367, 385, 414, 415, 416, 420, 423, 426, 428, 432, 438, 443, 447, 504, 505, 506, 536, 537, 540, 567, 569, 602, 604, 628, 660
Offset: 1

Views

Author

Raúl Mario Torres Silva, Jan 30 2019

Keywords

Comments

Let T be the set of permutations of nonnegative integers t such that t_i = i for all but a finite number of terms i.
The A306428 sequence enumerates the elements of T, hence we have a bijection f from T to the nonnegative integers.
The bijection f has the following properties: for any N > 0:
- if f(t) < N!, then t_i = i for any i >= N,
- this is consistent with the fact that there are N! permutations of (0..N-1),
- if f(t) + f(u) = N!-1, then t_i = u_{N-1-i} for i = 0..N-1,
- in other words, t and u, restricted to (0..N-1), are symmetrical permutations.
This sequence corresponds to the values f(t) of the permutations t in T such that t_i - t_j <> j - i for any distinct i and j.
Hence, for any n > 0 and N > 0:
- if a(n) < N!, then a(n) represents a permutation t of (0..N-1) such that the numbers t_i + i are distinct for i = 0..N-1; this corresponds to a configuration of N queens on an N X N board in which two queens do not attack each other if they are on the same northwest-southeast diagonal,
- this explains the expression of A099152 in the Formula section,
- also if a(n) = N! - 1 - a(m) for some m > 0, then a(n) represents a permutation t of (0..N-1) such that the numbers t_i + i are distinct for i = 0..N-1 and the numbers t_j - j are distinct for j = 0..N-1; this corresponds to a configuration of N nonattacking queens on an N X N board,
- this explains the expression of A000170 in the Formula section.

Examples

			For N = 6, there are 83 matrices in which the sums of the entries of each northeast-southwest diagonal are 0 or 1.
Also, for N = 6, there are 4 ways to place 6 nonattacking queens on a 6 X 6 board.
Finally, the solutions for N = 6 are 150, 296, 423 and 569 (positions within the ordered permutations, see A306428).
150 = (2,4,6,1,3,5);
O O O X O O
X O O O O O
O O O O X O
O X O O O O
O O O O O X
O O X O O O
296 = (3,6,2,5,1,4);
O O O O X O
O O X O O O
X O O O O O
O O O O O X
O O O X O O
O X O O O O
423 = (4,1,5,2,6,3);
O X O O O O
O O O X O O
O O O O O X
X O O O O O
O O X O O O
O O O O X O
569 = (5,3,1,6,4,2);
O O X O O O
O O O O O X
O X O O O O
O O O O X O
X O O O O O
O O O X O O

Links

Crossrefs

Cf. A000170, A099152, A306428

Formula

A099152(k) = Sum_{i > 0} [k! - 1 - a(i) >= 0] (with [] = Iverson bracket).
A000170(k) = Sum_{i > 0} [k! - 1 - a(i) belongs to {a(n)}].

A322859 The number of permutations p of {1,...,n} such that the numbers 2p(1)-1,...,2p(n)-n are all distinct.

Original entry on oeis.org

1, 1, 2, 4, 14, 52, 256, 1396, 9064, 62420, 500000, 4250180, 40738880, 410140060, 4572668112, 53214384548, 676739353112
Offset: 0

Views

Author

M. Farrokhi D. G., Dec 29 2018

Keywords

Comments

If p=(i,j) is a transposition on letters 1,...,n with 1 <= i < j <= n, then the numbers 2p(1)-1, ..., 2p(n)-n are all distinct if and only if either j >= 2i or j > (i+n)/2. It follows that the number b(n) of such permutations equals A000212(n)=floor(n^2/3).

Examples

			For n=4, the a(4)=14 permutations are (), (2,4), (2,3,4), (1,4), (1,4,3,2), (1,4,2,3), (1,4)(2,3), (1,2,4,3), (1,2)(3,4), (1,2,3,4), (1,3), (1,3,2), (1,3)(2,4), (1,3,2,4).

Crossrefs

Cf. A000212.
A099152 counts the permutations of {1,...,n} such that the numbers p(i)-i (or p(i)+i) are distinct for i=1,...,n.

Programs

  • GAP
    Number(Filtered(SymmetricGroup(n),p->Number(Unique(List([1..n],i->2*i^p-i)))=n));

Formula

Conjecture: n! ~ n^(1+o(1))*a(n).
Conjecture: (n-2)a(n-1) <= a(n) <= (n-1)a(n-1).
Conjecture: The polynomial a(1)+a(2)x+...+a(n)x^(n-1) is irreducible for all n. Indeed, it seems that the polynomials are irreducible for any permutation of coefficients except for n=7 where the exceptional permutations are (1,7,3,5,4,6) and (1,3,4,6,2).

Extensions

a(15)-a(16) from Bert Dobbelaere, Sep 18 2019
Previous Showing 11-19 of 19 results.

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