OFFSET
1,2
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
LINKS
Tilman Piesk, First 100 rows, flattened
Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)
FORMULA
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tilman Piesk, Dec 07 2013
STATUS
approved