A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

1, 2, 2, 5, 12, 6, 15, 64, 72, 24, 52, 350, 660, 480, 120, 203, 2024, 5670, 6720, 3600, 720, 877, 12460, 48552, 83160, 71400, 30240, 5040, 4140, 81638, 424536, 983808, 1201200, 806400, 282240, 40320

Offset: 1

Author

Tilman Piesk, Dec 07 2013

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of preferential arrangements with k levels of partitions of the set {1...n}.

2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and k runs of A's and E's (universal and existential quantifiers, compare runs of 0's ans 1's counted by A005811), but don't include a negator.

4*T(n,k) is the number of such formulas that may include an negator.

T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used. T(3,2) = 12: 1a|23b, 1b|23a, 13a|2b, 13b|2a, 12a|3b, 12b|3a, 1a|2a|3b, 1b|2b|3a, 1a|2b|3a, 1b|2a|3b, 1a|2b|3b, 1b|2a|3a. - Alois P. Heinz, Sep 01 2019

Examples

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
Compare descriptions of A083355 and A232598.
a(3,1)=5:
{1,2,3}
{1,2}{3}
{1,3}{2}
{2,3}{1}
{1}{2}{3}
a(3,2)=12:
{1,2}:{3}   {3}:{1,2}
{1,3}:{2}   {2}:{1,3}
{2,3}:{1}   {1}:{2,3}
{1}{2}:{3}   {3}:{1}{2}
{1}{3}:{2}   {2}:{1}{3}
{2}{3}:{1}   {1}:{2}{3}
a(3,3)=6:
{1}:{2}:{3}
{1}:{3}:{2}
{2}:{1}:{3}
{2}:{3}:{1}
{3}:{1}:{2}
{3}:{2}:{1}
Triangle begins:
       k = 1     2      3      4       5      6      7     8           sums
1          1                                                              1
2          2     2                                                        4
3          5    12      6                                                23
4         15    64     72     24                                        175
5         52   350    660    480     120                               1662
6        203  2024   5670   6720    3600    720                       18937
7        877 12460  48552  83160   71400  30240   5040               251729
8       4140 81638 424536 983808 1201200 806400 282240 40320        3824282
		

Links

• Tilman Piesk, First 100 rows, flattened
• Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)

Crossrefs

A008277 (Stirling2), A039810 (square of Stirling2), A000110 (Bell), A000142 (factorials), A083355 (row sums: number of preferential arrangements), A232598 (number of preferential arrangements by number of blocks).

Cf. A130191.

Formula

S2 = A008277 (Stirling numbers of the second kind).

(S2)^2 = A039810 (matrix square of S2).

T(n,k) = ((S2)^2)(n,k) * k! = Sum(k<=i<=n) [ S2(n,i) * S2(i,k) ] * k!.

T(n,1) = Bell(n) = A000110(n).

T(n,2) = A052896(n).

T(n,n) = n! = A000142(n).

T(n,n-1) = n!*(n-1) = A062119(n).