OFFSET
1,3
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<n) and before (i,j+1) (if j<n). The list is not always unique, however.
Given any set A={a_1<...<a_n} of real numbers with sumset A+A={s_1<...<s_k}, the ordered partition of S whose parts are the sets P_x = {(i,j): a_i+a_j = s_x}, with the parts ordered by P_1<P_2<...<P_k, gives an example of the type of ordered partition being counted.
Given any set A={a_1<...<a_n} of positive integers with sumset A+A={s_1<...<s_k}, the ordered partition of S whose parts are the sets P_x = {(i,j): a_i * a_j = s_x}, with the parts ordered by P_1<P_2<...<P_k, gives an example of the type of ordered partition being counted.
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
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..20
FORMULA
a(n) <= A000670(n*(n+1)/2).
EXAMPLE
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)
PROG
(PARI) \\ See PARI link in A374514 for program code.
vector(10, n, A376162(n)) \\ Andrew Howroyd, Sep 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin O'Bryant, Sep 12 2024
EXTENSIONS
a(7) onwards from Andrew Howroyd, Sep 15 2024
STATUS
approved