A323954 Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into connected subsequences of sizes > k, n >=1, 0 <= k < n.

1, 2, 1, 5, 1, 1, 12, 3, 1, 1, 27, 6, 1, 1, 1, 58, 12, 4, 1, 1, 1, 121, 22, 8, 1, 1, 1, 1, 248, 39, 13, 5, 1, 1, 1, 1, 503, 67, 22, 10, 1, 1, 1, 1, 1, 1014, 113, 36, 16, 6, 1, 1, 1, 1, 1, 2037, 188, 56, 23, 12, 1, 1, 1, 1, 1, 1, 4084, 310, 86, 35, 19, 7, 1, 1, 1, 1, 1, 1

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

T(n,k) = 1 - n + Sum_{i=1..floor(n/(k+1))} n*binomial(n-i*k-1, i-1)/i. - Andrew Howroyd, Jan 19 2023

Example

Triangle begins:
     1
     2    1
     5    1    1
    12    3    1    1
    27    6    1    1    1
    58   12    4    1    1    1
   121   22    8    1    1    1    1
   248   39   13    5    1    1    1    1
   503   67   22   10    1    1    1    1    1
  1014  113   36   16    6    1    1    1    1    1
  2037  188   56   23   12    1    1    1    1    1    1
  4084  310   86   35   19    7    1    1    1    1    1    1
Row 4 counts the following partitions:
  {{1234}}        {{1234}}    {{1234}}  {{1234}}
  {{1}{234}}      {{12}{34}}
  {{12}{34}}      {{14}{23}}
  {{123}{4}}
  {{124}{3}}
  {{134}{2}}
  {{14}{23}}
  {{1}{2}{34}}
  {{1}{23}{4}}
  {{12}{3}{4}}
  {{14}{2}{3}}
  {{1}{2}{3}{4}}

Mathematica

cycedsprop[n_, k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i, j]-1, n], {i, n}, {j, i+k, n+i-1}]];
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
Table[Length[spsu[cycedsprop[n, k], Range[n]]], {n, 12}, {k, 0, n-1}]

Prog

(PARI) T(n, k) = 1 - n + sum(i=1, n\(k+1), n*binomial(n-i*k-1, i-1)/i) \\ Andrew Howroyd, Jan 19 2023

Crossrefs

Column k = 0 is A000325. Column k = 1 is A066982. Column k = 2 is A323951. Column k = 3 is A306351.

Cf. A001610, A001680, A005251, A323950, A323951, A323952, A323953.

Sequence in context: A369526 A206563 A299779 * A143983 A282988 A113767

Adjacent sequences: A323951 A323952 A323953 * A323955 A323956 A323957

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 10 2019

Status

approved