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

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, 1, 14832, 4376, 696, 118, 24, 6, 2, 1, 133497, 35620, 4923, 714, 120, 24, 6, 2, 1, 1334960, 324965, 39612, 5016, 718, 120, 24, 6, 2, 1, 14684571, 3285270, 357900, 40200, 5034, 720, 120, 24, 6, 2, 1

Offset: 1

Author

Seiichi Manyama, Feb 21 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
      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, 1;
  14832, 4376, 696, 118, 24, 6, 2, 1;
		

Links

• R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Columns k=1..4 give A000240, A370524, A370525, A369098.

Cf. A008290, A177248, A177250, A177252.

Programs

PARI

T(n, k) = sum(j=0, n\k-1, (-1)^j*(n-(k-1)*(j+1))!/j!);

Formula

G.f. of column k: Sum_{j>=1} j! * x^(j+k-1) / (1+x^k)^(j+1).

T(n,k) = Sum_{j=0..floor(n/k)-1} (-1)^j * (n-(k-1)*(j+1))! / j!.