A376162 Number of ordered partitions of S={(i,j):1 <= i , j <= n} where for every i and j the pairs (i+1,j) and (i,j+1) are in a later part than the part containing the pair (i,j), and the pairs (i,j), (j,i) are in the same part.

1, 1, 3, 39, 2905, 1538369, 6904262355, 304662492057063, 150347237334006997801, 929721796071361437087789041, 79773595676787229793797978773561927, 104165556509336140832819242491033872033130063, 2252283824141388832759484222915451435885285752729087857

Offset: 1

Author

Kevin O'Bryant, Sep 12 2024

Keywords

• nonn

Comments

Ordered partitions are also called weak orderings.

Any such ordered partition can be written as a list of pairs (i,j) with 1 <= i <= j <= n, with either "=" or "<" between each pair, and so that (i,j) appears in the list before (i+1,j) (if i

Given any set A={a_1<...

Given any set A={a_1<...

Equivalently, a(n) is the number of n X n symmetric matrices whose values cover an initial interval of positive integers and whose rows have values which are strictly increasing. - Andrew Howroyd, Sep 15 2024

Examples

			For n=2 the a(2)=1 ordered partition is {(1,1)}<{(2,1),(1,2)}<{(2,2)}. We can encode this as 11<12<22, writing "ij" for the pair (i,j).
For n=3 one of the a(3)=3 ordered partitions is {(1,1)}<{(1,2),(2,1)}<{(1,3),(3,1),(2,2)}<{(2,3),(3,2)}<{(3,3)}, which is encoded as either 11<12<13=22<23<33 or 11<12<22=13<23<33. The other two ordered partitions can be encoded as 11<12<22<13<23<33 and 11<12<13<22<23<33.
From _Andrew Howroyd_, Sep 15 2024: (Start)
The a(3) = 3 symmetric matrices are:
    [1 2 3]   [1 2 3]   [1 2 4]
    [2 3 4]   [2 4 5]   [2 3 5]
    [3 4 5]   [3 5 6]   [4 5 6]
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..20

Crossrefs

Cf. A000670, A006541, A003121, A374514.

Programs

PARI

\\ See PARI link in A374514 for program code.
vector(10, n, A376162(n)) \\ Andrew Howroyd, Sep 16 2024

Formula

a(n) <= A000670(n*(n+1)/2).

Extensions

a(7) onwards from Andrew Howroyd, Sep 15 2024

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