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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

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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

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).

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.

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

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.

Sequence in context: A208258 A208872 A208581 * A248664 A143195 A192799

Adjacent sequences: A233354 A233355 A233356 * A233358 A233359 A233360

KEYWORD

nonn,tabl

AUTHOR

Tilman Piesk, Dec 07 2013

STATUS

approved