A130760 -id:A130760 - cp's OEIS frontend

A211351 Refined triangle A091867: T(n,k) is the number of noncrossing partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

1, 1, 1, 1, 3, 1, 1, 6, 4, 2, 1, 1, 10, 10, 10, 5, 5, 1, 1, 15, 20, 30, 15, 30, 6, 5, 6, 3, 1, 1, 21, 35, 70, 35, 105, 21, 35, 42, 21, 7, 21, 7, 7, 1, 1, 28, 56, 140, 70, 280, 56, 140, 168, 84, 28, 168, 56, 56, 8, 14, 28, 28, 8, 8, 4, 1

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,5,14,42,132... (Catalan numbers A000108)
Row maxima: 1,1,3,6,10,30,105,280,756,2520,6930,18480 (A130760)
Distinct entries per row: 1,1,2,4,3,7,7,11,12,18,18,30
Rightmost columns are those from Pascal's triangle A007318 without the second one (i.e. triangle A184049). The other columns - (always?) without a 1 at the top - are multiples of these columns from Pascal's triangle; so actually only the top elements of each column are needed to calculate the other entries; these top elements are in A211361.

Tilman Piesk, Rows n=1..12 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A091867, A000108, A130760, A184049, A211361.

A247651 Maximum number of binary strings of length 2n obtained from a partition of n.

Original entry on oeis.org

1, 2, 3, 12, 30, 60, 210, 840, 2520, 7560, 27720, 83160, 240240, 840840, 2702700, 10810800, 36756720, 122522400, 465585120, 1551950400, 4888643760, 19554575040, 74959204320, 257002986240, 936990054000, 3480248772000, 11745839605500, 40477970332800, 146732642456400, 524045151630000

Offset: 0

Andrei Cretu and Yuri Dimitrov, Oct 03 2014

nonn

The number of different binary strings of length 2n that can be constructed with an equal number (n) of 0's and 1's, based on a given partition of the 0's (or 1's) into uninterrupted runs, can be written as Nseq(n,partition)=(n+1)!/(Prod_j(m_j!)(n-m+1)!) where m is the number of partition members (total number of runs of 0's or 1's); and m_j is the multiplicity of runs of length j of 0's (or 1's) (j=positive integer).
The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
Prod_j(m_j!)(n-m+1)! becomes n! at the extremes (finest partition of n, m=n -- coarsest partition of n, m=1). Nseq (n,partition) is in that sense a relative measure of the complexity of the partition and the associated binary strings. a(n) is the number of strings obtained based on the partition of n that maximizes Nseq(n,partition).

			n=0 gives the empty string.
n=1 and the only possible partition generate 01 and 10.
For n=2, both possible partitions generate 3 strings (0011,0110 and 1100, and respectively 0101, 1001 and 1010, based on runs of 1's).
For n=3, the optimal partition is {1,2}, generating 12 strings (based on runs of 1's: 001011, 001101, 010011, 010110, 011001, 011010, 100011, 100110, 101100, 110001, 110010, 110100).

Cf. A025560, A130760.

nseq[p_]:=FactorialPower[Total[p]+1,Length[p]]/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]];
a[n_]:=Max[Map[nseq,IntegerPartitions[n]]]
Table[a[n],{n,0,20}] (* after A130670 *)

a(n) = (n+1)*A130760(n).

a(n) = Max[(n+1)!/(Prod_j(m_j!)(n-m+1)!)] over all partitions of n.

More terms from Michel Marcus, May 19 2025

cp's OEIS Frontend

A211351 Refined triangle A091867: T(n,k) is the number of noncrossing partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

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A247651 Maximum number of binary strings of length 2n obtained from a partition of n.

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