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

%I A233357 #21 Apr 07 2020 22:00:07
%S A233357 1,2,2,5,12,6,15,64,72,24,52,350,660,480,120,203,2024,5670,6720,3600,
%T A233357 720,877,12460,48552,83160,71400,30240,5040,4140,81638,424536,983808,
%U A233357 1201200,806400,282240,40320
%N A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!
%C A233357 T(n,k) is the number of preferential arrangements with k levels of partitions of the set {1...n}.
%C A233357 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 A233357 4*T(n,k) is the number of such formulas that may include an negator.
%C A233357 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 A233357 Tilman Piesk, <a href="/A233357/b233357.txt">First 100 rows, flattened</a>
%H A233357 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Preferential_arrangements_of_set_partitions">Preferential arrangements of set partitions</a> (Wikiversity)
%F A233357 S2 = A008277 (Stirling numbers of the second kind).
%F A233357 (S2)^2 = A039810 (matrix square of S2).
%F A233357 T(n,k) = ((S2)^2)(n,k) * k! = Sum(k<=i<=n) [ S2(n,i) * S2(i,k) ] * k!.
%F A233357 T(n,1) = Bell(n) = A000110(n).
%F A233357 T(n,2) = A052896(n).
%F A233357 T(n,n) = n! = A000142(n).
%F A233357 T(n,n-1) = n!*(n-1) = A062119(n).
%e A233357 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 A233357 Compare descriptions of A083355 and A232598.
%e A233357 a(3,1)=5:
%e A233357 {1,2,3}
%e A233357 {1,2}{3}
%e A233357 {1,3}{2}
%e A233357 {2,3}{1}
%e A233357 {1}{2}{3}
%e A233357 a(3,2)=12:
%e A233357 {1,2}:{3}   {3}:{1,2}
%e A233357 {1,3}:{2}   {2}:{1,3}
%e A233357 {2,3}:{1}   {1}:{2,3}
%e A233357 {1}{2}:{3}   {3}:{1}{2}
%e A233357 {1}{3}:{2}   {2}:{1}{3}
%e A233357 {2}{3}:{1}   {1}:{2}{3}
%e A233357 a(3,3)=6:
%e A233357 {1}:{2}:{3}
%e A233357 {1}:{3}:{2}
%e A233357 {2}:{1}:{3}
%e A233357 {2}:{3}:{1}
%e A233357 {3}:{1}:{2}
%e A233357 {3}:{2}:{1}
%e A233357 Triangle begins:
%e A233357        k = 1     2      3      4       5      6      7     8           sums
%e A233357 1          1                                                              1
%e A233357 2          2     2                                                        4
%e A233357 3          5    12      6                                                23
%e A233357 4         15    64     72     24                                        175
%e A233357 5         52   350    660    480     120                               1662
%e A233357 6        203  2024   5670   6720    3600    720                       18937
%e A233357 7        877 12460  48552  83160   71400  30240   5040               251729
%e A233357 8       4140 81638 424536 983808 1201200 806400 282240 40320        3824282
%Y A233357 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 A233357 Cf. A130191.
%K A233357 nonn,tabl
%O A233357 1,2
%A A233357 _Tilman Piesk_, Dec 07 2013