A213338 -id:A213338 - cp's OEIS frontend

A214337 Triangle read by rows: T(n,k) = number of rooted maps with n vertices and k faces on a non-orientable surface of type 3/2 (0 <= k <= n).

0, 0, 41, 0, 690, 16925, 0, 7150, 237652, 4306778, 0, 58760, 2518957, 56864524, 910734615, 0, 420182, 22417804, 613687758, 11675167470, 174833737848

Offset: 0

N. J. A. Sloane, Jul 27 2012

			Triangle begins:
  0;
  0,     41;
  0,    690,    16925;
  0,   7150,   237652,   4306778;
  0,  58760,  2518957,  56864524,   910734615;
  0, 420182, 22417804, 613687758, 11675167470, 174833737848;
  ...

Didier Arquès and Alain Giorgetti, Counting rooted maps on a surface, Theoret. Comput. Sci. 234 (2000), no. 1-2, 255--272. MR1745078 (2001f:05078).

Diagonals give A118448, A214335, A213336, A213338.

Cf. A214806.

A213270 Costas arrays such that the corresponding permutation is an involution.

Original entry on oeis.org

1, 2, 2, 2, 4, 10, 20, 18, 20, 28, 36, 34, 50, 46, 62, 40, 38, 20, 12, 8, 16, 10, 20, 0, 4, 4, 14, 0, 10

Offset: 1

Joerg Arndt, Jun 08 2012

Self-inverse permutations such that each row in the difference table consists of pairwise distinct elements (see example).

			The permutation (4, 7, 9, 1, 6, 5, 2, 8, 3) is an involution and corresponds to a Costas array:
   4  7  9  1  6  5  2  8  3  (Permutation: p(1), p(2), p(3), ..., p(n) )
   3  2 -8  5 -1 -3  6 -5     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
   5 -6 -3  4 -4  3  1        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -3 -1 -4  1  2 -2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
   2 -2 -7  7 -3              ( etc. )
   1 -5 -1  2
  -2  1 -6
   4 -4
  -1

Scott Rickard, costasarrays.org (information and papers about Costas arrays).

Cf. A008404 (Costas arrays), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).

A213271 Costas arrays such that the corresponding permutation is a derangement.

Original entry on oeis.org

0, 1, 2, 2, 18, 42, 66, 168, 300, 910, 1882, 3192, 5320, 7166, 8346, 9042, 7760, 6668, 4620, 2822, 1528, 942, 282, 92, 32, 22, 88, 256, 24

Offset: 1

Joerg Arndt, Jun 08 2012

Fixed-point free permutations such that each row in the difference table consists of pairwise distinct elements (see example).

			The permutation (9, 8, 1, 6, 3, 7, 2, 4, 5) is a derangement and corresponds to a Costas array:
   9  8  1  6  3  7  2  4  5  (Permutation: p(1), p(2), p(3), ..., p(n) )
  -1 -7  5 -3  4 -5  2  1     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
  -8 -2  2  1 -1 -3  3        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -3 -5  6 -4  1 -2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
  -6 -1  1 -2  2              ( etc. )
  -2 -6  3 -1
  -7 -4  4
  -5 -3
  -4

Scott Rickard, costasarrays.org (information and papers about Costas arrays).

Cf. A008404 (Costas arrays), A213270 (Costas arrays that are involutions), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).

A213339 Costas arrays such that the corresponding permutation is connected.

Original entry on oeis.org

1, 1, 2, 6, 26, 80, 152, 348, 628, 1868, 3870, 7014, 11788, 15746, 18388, 19820, 17218, 14344, 9844, 6238, 3430, 1968, 814, 184, 84, 52, 190, 656, 132

Offset: 1

Joerg Arndt, Jun 09 2012

Scott Rickard, costasarrays.org (information and papers about Costas arrays).

Cf. A008404 (Costas arrays), A003319 (connected permutations), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic).

A213272 Costas arrays such that the terms in each row of the difference table are unique modulo n.

Original entry on oeis.org

1, 2, 0, 8, 0, 12, 0, 0, 0, 40, 0, 48, 0, 0, 0, 128, 0, 108, 0, 0, 0, 220, 0, 0, 0, 0, 0, 336, 0

Offset: 1

Joerg Arndt, Jun 08 2012

Permutations of n elements such that each row in the difference table consists of pairwise distinct elements, even when taken modulo n (see example).
For n<=29 the nonzero terms a(n) appear for n in A006093 (primes minus 1) and a(n)=A002618(n) (n*phi(n)); omitting the zeros we obtain A104039 (number of primitive roots modulo (p(n))^2, where p(n) is n-th prime).
A002618(n) divides a(n) for all n, since (treating elements as integers modulo n) adding or subtracting a constant from each element or multiplying each element by an integer coprime to n preserves distinctness of all values modulo n. - Charlie Neder, May 26 2019

			The permutation (10, 9, 2, 8, 6, 1, 3, 7, 4, 5) corresponds to a Costas array:
  10  9  2  8  6  1  3  7  4  5  (Permutation: p(1), p(2), p(3), ..., p(n) )
  -1 -7  6 -2 -5  2  4 -3  1     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
  -8 -1  4 -7 -3  6  1 -2        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -2 -3 -1 -5  1  3  2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
  -4 -8  1 -1 -2  4              ( etc. )
  -9 -6  5 -4 -1
  -7 -2  2 -3
  -3 -5  3
  -6 -4
  -5
The values in each row are unique also modulo n=10:
  10 9 2 8 6 1 3 7 4 5  (Permutation: p(1), p(2), p(3), ..., p(n) )
   9 3 6 8 5 2 4 7 1    (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
   2 9 4 3 7 6 1 8      (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
   8 7 9 5 1 3 2        (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
   6 2 1 9 8 4          ( etc. )
   1 4 5 6 9
   3 8 2 7
   7 5 3
   4 6
   5

Scott Rickard, costasarrays.org (information and papers about Costas arrays). [broken link?]
Wikipedia, Costas array.

Cf. A008404 (Costas arrays), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).

cp's OEIS Frontend

A214337 Triangle read by rows: T(n,k) = number of rooted maps with n vertices and k faces on a non-orientable surface of type 3/2 (0 <= k <= n).

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A213270 Costas arrays such that the corresponding permutation is an involution.

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A213271 Costas arrays such that the corresponding permutation is a derangement.

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A213339 Costas arrays such that the corresponding permutation is connected.

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A213272 Costas arrays such that the terms in each row of the difference table are unique modulo n.

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