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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).
3
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
1,4
LINKS
R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.
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!.
EXAMPLE
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;
PROG
(PARI) T(n, k) = sum(j=0, n\k-1, (-1)^j*(n-(k-1)*(j+1))!/j!);
CROSSREFS
Columns k=1..4 give A000240, A370524, A370525, A369098.
Sequence in context: A158474 A090452 A305538 * A193924 A110439 A327917
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 21 2024
STATUS
approved