A197126 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of cliques of size k in all partitions of n.

1, 1, 1, 3, 0, 1, 4, 2, 0, 1, 8, 2, 1, 0, 1, 11, 4, 2, 1, 0, 1, 19, 5, 3, 1, 1, 0, 1, 26, 10, 3, 3, 1, 1, 0, 1, 41, 11, 7, 3, 2, 1, 1, 0, 1, 56, 20, 8, 5, 3, 2, 1, 1, 0, 1, 83, 25, 13, 6, 5, 2, 2, 1, 1, 0, 1, 112, 38, 17, 11, 5, 5, 2, 2, 1, 1, 0, 1, 160, 49, 25, 13, 9, 5, 4, 2, 2, 1, 1, 0, 1

Offset

1,4

Comments

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

Links

Alois P. Heinz, Rows n = 1..141, flattened

Abdulaziz M. Alanazi and Augustine O. Munagi, On partition configurations of Andrews-Deutsch, Integers 17 (2017), #A7.

Formula

G.f. of column k: (x^k/(1-x^k)-x^(k+1)/(1-x^(k+1)))/Product_{j>0}(1-x^j).

Column k is asymptotic to exp(Pi*sqrt(2*n/3)) / (k*(k+1)*Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, May 24 2018

Example

T(4,1) = 4: [1,1,(2)], [(1),(3)], [(4)].
T(8,3) = 3: [1,1,(2,2,2)], [(1,1,1),2,3], [(1,1,1),5].
T(12,4) = 11: [(1,1,1,1),(2,2,2,2)], [1,(2,2,2,2),3], [(1,1,1,1),2,3,3], [(3,3,3,3)], [(1,1,1,1),2,2,4], [(2,2,2,2),4], [(1,1,1,1),4,4], [(1,1,1,1),3,5], [(1,1,1,1),2,6], [(1,1,1,1),8].  Here the first partition contains 2 cliques.
Triangle begins:
   1;
   1,  1;
   3,  0, 1;
   4,  2, 0, 1;
   8,  2, 1, 0, 1;
  11,  4, 2, 1, 0, 1;
  19,  5, 3, 1, 1, 0, 1;
  26, 10, 3, 3, 1, 1, 0, 1;
  ...

Maple

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..20);

Mathematica

Table[CoefficientList[ 1/q* Tr[Flatten[q^Map[Length, Split /@ IntegerPartitions[n], {2}]]], q], {n, 24}] (* Wouter Meeussen, Apr 21 2012 *)
b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == k, l + {0, l[[1]]}, l]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Crossrefs

Columns k=1-10 give: A024786(n+1), A116646, A117524, A222704, A222705, A222706, A222707, A222708, A222709, A222710.

Row sums give: A000070(n-1). Diagonal gives: A000012. Limit of reversed rows: T(2*n+1,n+1) = A002865(n).

Cf. A213180.

Sequence in context: A245120 A226912 A177330 * A256987 A048963 A119458

Adjacent sequences: A197123 A197124 A197125 * A197127 A197128 A197129

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 10 2011

Status

approved