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A370527 Triangle read by rows: T(n,k) = number of permutations of [n] having exactly one adjacent k-cycle. (n>=1, 1<=k<=n).

%I A370527 #19 Feb 25 2024 00:11:22
%S A370527 1,0,1,3,2,1,8,4,2,1,45,18,6,2,1,264,99,22,6,2,1,1855,612,114,24,6,2,
%T A370527 1,14832,4376,696,118,24,6,2,1,133497,35620,4923,714,120,24,6,2,1,
%U A370527 1334960,324965,39612,5016,718,120,24,6,2,1,14684571,3285270,357900,40200,5034,720,120,24,6,2,1
%N A370527 Triangle read by rows: T(n,k) = number of permutations of [n] having exactly one adjacent k-cycle. (n>=1, 1<=k<=n).
%H A370527 R. A. Brualdi and Emeric Deutsch, <a href="http://arxiv.org/abs/1005.0781">Adjacent q-cycles in permutations</a>, arXiv:1005.0781 [math.CO], 2010.
%F A370527 G.f. of column k: Sum_{j>=1} j! * x^(j+k-1) / (1+x^k)^(j+1).
%F A370527 T(n,k) = Sum_{j=0..floor(n/k)-1} (-1)^j * (n-(k-1)*(j+1))! / j!.
%e A370527 Triangle starts:
%e A370527       1;
%e A370527       0,    1;
%e A370527       3,    2,   1;
%e A370527       8,    4,   2,   1;
%e A370527      45,   18,   6,   2,  1;
%e A370527     264,   99,  22,   6,  2, 1;
%e A370527    1855,  612, 114,  24,  6, 2, 1;
%e A370527   14832, 4376, 696, 118, 24, 6, 2, 1;
%o A370527 (PARI) T(n, k) = sum(j=0, n\k-1, (-1)^j*(n-(k-1)*(j+1))!/j!);
%Y A370527 Columns k=1..4 give A000240, A370524, A370525, A369098.
%Y A370527 Cf. A008290, A177248, A177250, A177252.
%K A370527 nonn,tabl
%O A370527 1,4
%A A370527 _Seiichi Manyama_, Feb 21 2024