cp's OEIS Frontend

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.

%I A341439 #28 Jun 01 2022 07:59:35
%S A341439 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,
%T A341439 2,13,80,579,4738,0,1,1,4,44,82,579,4740,43387,0,0,1,2,13,80,579,4738,
%U A341439 43387,439792,0,1,2,9,13,265,579,4752,43390,439794,4890741
%N 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.
%C A341439 The recurrence for the second row comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.
%H A341439 D. Zeilberger, <a href="https://arxiv.org/abs/1401.1089">Automatic Enumeration of Generalized Menage Numbers</a>, arXiv preprint arXiv:1401.1089 [math.CO], 2014.
%F A341439 T(n,n) = A000166(n) for n >= 1.
%F A341439 T(1,k) = A000179(k).
%F A341439 T(k-1,k) = A000179(k) for k >= 2.
%F A341439 T(n,k) = T(n+k, k).
%F A341439 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.
%F A341439 T(n,k) = A277256(gcd(n,k),k/gcd(n,k)). - _Pontus von Brömssen_, May 31 2022
%e A341439 Table begins:
%e A341439 n\k | 1 2 3 4  5   6    7     8
%e A341439 ----+--------------------------
%e A341439   1 | 0 0 1 2 13  80  579  4738
%e A341439   2 | 0 1 1 4 13  82  579  4740
%e A341439   3 | 0 0 2 2 13  80  579  4738
%e A341439   4 | 0 1 1 9 13  82  579  4752
%e A341439   5 | 0 0 1 2 44  80  579  4738
%e A341439   6 | 0 1 2 4 13 265  579  4740
%e A341439   7 | 0 0 1 2 13  80 1854  4738
%e A341439   8 | 0 1 1 9 13  82  579 14833
%Y A341439 Cf. A000166, A000179, A277256, A354408.
%K A341439 nonn,tabl
%O A341439 1,10
%A A341439 _Peter Kagey_, Feb 11 2021