cp's OEIS Frontend

A232598 T(n,k) = Stirling2(n,k) * OrderedBell(k).

%I A232598 #32 Jun 02 2025 08:50:34
%S A232598 1,1,3,1,9,13,1,21,78,75,1,45,325,750,541,1,93,1170,4875,8115,4683,1,
%T A232598 189,3913,26250,75740,98343,47293,1,381,12558,127575,568050,1245678,
%U A232598 1324204,545835,1,765,39325,582750,3760491,12391218,21849366,19650060,7087261
%N A232598 T(n,k) = Stirling2(n,k) * OrderedBell(k).
%C A232598 T(n,k) is the number of preferential arrangements of the k-part partitions of the set {1...n}.
%C A232598 2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and use k variables, but don't include a negator.
%C A232598 4*T(n,k) is the number of such formulas that may include an negator.
%C A232598 The entries T(n,n) are A000670(n), i.e. the ordered Bell numbers.
%H A232598 Tilman Piesk, <a href="/A232598/b232598.txt">First 100 rows, flattened</a>
%H A232598 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Preferential_arrangements_of_set_partitions">Preferential arrangements of set partitions</a> (Wikiversity)
%F A232598 T(n,k) = A008277(n,k) * A000670(k).
%F A232598 T(n,n) = A000670(n).
%F A232598 T(n,2) = A068156(n-1).
%F A232598 From _Peter Bala_, Nov 27 2013: (Start)
%F A232598 E.g.f.: 1/( 2 - exp(x*(exp(t) - 1)) ) = 1 + x*t + (x + 3*x^2)*t^2/2! + (x + 9*x^2 + 13*x^3)*t^3/3! + ....
%F A232598 Recurrence equation (for entries not on main diagonal): (n - k)*T(n,k) = C(n,1)*T(n-1,k) - C(n,2)*T(n-2,k) + C(n,3)*T(n-3,k) - ... (End)
%e A232598 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 A232598 Compare descriptions of A083355 and A233357.
%e A232598 a(3,1) = 1:
%e A232598 {1,2,3}
%e A232598 a(3,2) = 9:
%e A232598 {1,2}{3}
%e A232598 {1,3}{2}
%e A232598 {2,3}{1}
%e A232598 {1,2}:{3}   {3}:{1,2}
%e A232598 {1,3}:{2}   {2}:{1,3}
%e A232598 {2,3}:{1}   {1}:{2,3}
%e A232598 a(3,3) = 13:
%e A232598 {1}{2}{3}
%e A232598 {1}{2}:{3}   {3}:{1}{2}
%e A232598 {1}{3}:{2}   {2}:{1}{3}
%e A232598 {2}{3}:{1}   {1}:{2}{3}
%e A232598 {1}:{2}:{3}
%e A232598 {1}:{3}:{2}
%e A232598 {2}:{1}:{3}
%e A232598 {2}:{3}:{1}
%e A232598 {3}:{1}:{2}
%e A232598 {3}:{2}:{1}
%e A232598 Triangle begins:
%e A232598      k = 1   2     3      4      5       6       7      8          sums
%e A232598 n
%e A232598 1        1                                                            1
%e A232598 2        1   3                                                        4
%e A232598 3        1   9    13                                                 23
%e A232598 4        1  21    78     75                                         175
%e A232598 5        1  45   325    750    541                                 1662
%e A232598 6        1  93  1170   4875   8115    4683                        18937
%e A232598 7        1 189  3913  26250  75740   98343   47293               251729
%e A232598 8        1 381 12558 127575 568050 1245678 1324204 545835       3824282
%Y A232598 A008277 (Stirling2), A000670 (ordered Bell), A068156 (column k=2), A083355 (row sums: number of preferential arrangements), A233357 (number of preferential arrangements by number of levels).
%K A232598 nonn,tabl
%O A232598 1,3
%A A232598 _Tilman Piesk_, Nov 26 2013