cp's OEIS Frontend

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.

%I A323954 #17 Jan 19 2023 12:23:14
%S A323954 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,
%T A323954 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,
%U A323954 2037,188,56,23,12,1,1,1,1,1,1,4084,310,86,35,19,7,1,1,1,1,1,1
%N 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.
%H A323954 Andrew Howroyd, <a href="/A323954/b323954.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A323954 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
%e A323954 Triangle begins:
%e A323954      1
%e A323954      2    1
%e A323954      5    1    1
%e A323954     12    3    1    1
%e A323954     27    6    1    1    1
%e A323954     58   12    4    1    1    1
%e A323954    121   22    8    1    1    1    1
%e A323954    248   39   13    5    1    1    1    1
%e A323954    503   67   22   10    1    1    1    1    1
%e A323954   1014  113   36   16    6    1    1    1    1    1
%e A323954   2037  188   56   23   12    1    1    1    1    1    1
%e A323954   4084  310   86   35   19    7    1    1    1    1    1    1
%e A323954 Row 4 counts the following partitions:
%e A323954   {{1234}}        {{1234}}    {{1234}}  {{1234}}
%e A323954   {{1}{234}}      {{12}{34}}
%e A323954   {{12}{34}}      {{14}{23}}
%e A323954   {{123}{4}}
%e A323954   {{124}{3}}
%e A323954   {{134}{2}}
%e A323954   {{14}{23}}
%e A323954   {{1}{2}{34}}
%e A323954   {{1}{23}{4}}
%e A323954   {{12}{3}{4}}
%e A323954   {{14}{2}{3}}
%e A323954   {{1}{2}{3}{4}}
%t A323954 cycedsprop[n_,k_]:=Union[Sort/@Join@@Table[1+Mod[Range[i,j]-1,n],{i,n},{j,i+k,n+i-1}]];
%t A323954 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A323954 Table[Length[spsu[cycedsprop[n,k],Range[n]]],{n,12},{k,0,n-1}]
%o A323954 (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
%Y A323954 Column k = 0 is A000325. Column k = 1 is A066982. Column k = 2 is A323951. Column k = 3 is A306351.
%Y A323954 Cf. A001610, A001680, A005251, A323950, A323951, A323952, A323953.
%K A323954 nonn,tabl
%O A323954 1,2
%A A323954 _Gus Wiseman_, Feb 10 2019