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

%I #21 Apr 07 2020 22:00:07

%S 1,2,2,5,12,6,15,64,72,24,52,350,660,480,120,203,2024,5670,6720,3600,

%T 720,877,12460,48552,83160,71400,30240,5040,4140,81638,424536,983808,

%U 1201200,806400,282240,40320

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

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

%C 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.

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

%C 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

%H Tilman Piesk, <a href="/A233357/b233357.txt">First 100 rows, flattened</a>

%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Preferential_arrangements_of_set_partitions">Preferential arrangements of set partitions</a> (Wikiversity)

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

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

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

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

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

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

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

%e 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.

%e Compare descriptions of A083355 and A232598.

%e a(3,1)=5:

%e {1,2,3}

%e {1,2}{3}

%e {1,3}{2}

%e {2,3}{1}

%e {1}{2}{3}

%e a(3,2)=12:

%e {1,2}:{3} {3}:{1,2}

%e {1,3}:{2} {2}:{1,3}

%e {2,3}:{1} {1}:{2,3}

%e {1}{2}:{3} {3}:{1}{2}

%e {1}{3}:{2} {2}:{1}{3}

%e {2}{3}:{1} {1}:{2}{3}

%e a(3,3)=6:

%e {1}:{2}:{3}

%e {1}:{3}:{2}

%e {2}:{1}:{3}

%e {2}:{3}:{1}

%e {3}:{1}:{2}

%e {3}:{2}:{1}

%e Triangle begins:

%e k = 1 2 3 4 5 6 7 8 sums

%e 1 1 1

%e 2 2 2 4

%e 3 5 12 6 23

%e 4 15 64 72 24 175

%e 5 52 350 660 480 120 1662

%e 6 203 2024 5670 6720 3600 720 18937

%e 7 877 12460 48552 83160 71400 30240 5040 251729

%e 8 4140 81638 424536 983808 1201200 806400 282240 40320 3824282

%Y 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).

%Y Cf. A130191.

%K nonn,tabl

%O 1,2

%A _Tilman Piesk_, Dec 07 2013