A354408 -id:A354408 - cp's OEIS frontend

A354409 Maximum value in the n-th row of A354408.

0, 1, 4, 13, 82, 579, 4752, 43390, 440192, 4890741, 59245120, 775596313, 10930514688, 164806652728, 2649865335040, 45226435601207, 817154768973824, 15574618910994665, 312426715251262464, 6577619798222863696, 145060238642780180480, 3343382818203784146955

Offset: 2

Peter Kagey, May 25 2022

nonn

The minimum value appears to be given by A000179.
The differences between this sequence and A000179 begin 0, 0, 2, 0, 2, 0, 14, 3, 400, 0, 28478, ...
Conjecture: a(n) = A000179(n) if and only if n is prime.

Pontus von Brömssen, Table of n, a(n) for n = 2..450

Cf. A000179, A032742, A277256, A354408.

Conjecture: a(n) = A354408(n, A032742(n)) for n != 6. - Pontus von Brömssen, May 31 2022

a(13)-a(23) from Pontus von Brömssen, May 31 2022

A341439 Table of generalized ménage numbers read by antidiagonals upward: T(n,k) is the number of permutations pi in S_k such that pi(i) != i, i+n (mod k) for all i; n, k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 2, 4, 13, 0, 0, 1, 2, 13, 80, 0, 1, 1, 9, 13, 82, 579, 0, 0, 2, 2, 13, 80, 579, 4738, 0, 1, 1, 4, 44, 82, 579, 4740, 43387, 0, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 0, 1, 2, 9, 13, 265, 579, 4752, 43390, 439794, 4890741

Offset: 1

Peter Kagey, Feb 11 2021

The recurrence for the second row comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.

			Table begins:
n\k | 1 2 3 4  5   6    7     8
----+--------------------------
  1 | 0 0 1 2 13  80  579  4738
  2 | 0 1 1 4 13  82  579  4740
  3 | 0 0 2 2 13  80  579  4738
  4 | 0 1 1 9 13  82  579  4752
  5 | 0 0 1 2 44  80  579  4738
  6 | 0 1 2 4 13 265  579  4740
  7 | 0 0 1 2 13  80 1854  4738
  8 | 0 1 1 9 13  82  579 14833

D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000166, A000179, A277256, A354408.

T(n,n) = A000166(n) for n >= 1.
T(1,k) = A000179(k).
T(k-1,k) = A000179(k) for k >= 2.
T(n,k) = T(n+k, k).
T(2,k) = k*T(2,k-1) + 3*T(2,k-2) + (-2*k+6)*T(2,k-3) - 3*T(2,k-4) + (k-6)*T(2,k-5) + T(2,k-6) for k > 8.
T(n,k) = A277256(gcd(n,k),k/gcd(n,k)). - Pontus von Brömssen, May 31 2022

A354152 a(n) is the number of permutations pi in S_n such that pi(i) - i != 1 (mod n) and pi(i) - i != -1 (mod n) for all i.

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 82, 579, 4740, 43387, 439794, 4890741, 59216644, 775596313, 10927434466, 164806435783, 2649391469060, 45226435601207, 817056406224418, 15574618910994665, 312400218671253764, 6577618644576902053, 145051250421230224306, 3343382818203784146955

Offset: 0

Peter Kagey, May 27 2022

nonn

For n > 1, this is the number of ways of rearranging guests sitting at a circular table such that a guest may stay in the same seat, but cannot move exactly one seat to their left or right.

The recurrence comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.

			For n = 5, the a(5) = 13 permutations are 12345, 12543, 14325, 14523, 15342, 32145, 34125, 34512, 35142, 42315, 42513, 45123, and 45312.
The first letter is never 2 or 5, the second letter is never 1 or 3, the third letter is never 2 or 4, the fourth letter is never 3 or 5 and the fifth letter is never 1 or 4.

Peter Kagey, Table of n, a(n) for n = 0..400
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000179, A341439, A354408.

a(n) = n*a(n-1) + 3*a(n-2) + (-2n+6)*a(n-3) - 3*a(n-4) + (n-6)*a(n-5) + a(n-6) for n > 8.
a(2k+1) = A000179(2k+1) for k > 1.
Conjecture: a(2k) = A000179(2k) + 2 for k > 1.

cp's OEIS Frontend

A354409 Maximum value in the n-th row of A354408.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A341439 Table of generalized ménage numbers read by antidiagonals upward: T(n,k) is the number of permutations pi in S_k such that pi(i) != i, i+n (mod k) for all i; n, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A354152 a(n) is the number of permutations pi in S_n such that pi(i) - i != 1 (mod n) and pi(i) - i != -1 (mod n) for all i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula