cp's OEIS Frontend

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.

%I A376162 #24 Sep 16 2024 23:59:28
%S A376162 1,1,3,39,2905,1538369,6904262355,304662492057063,
%T A376162 150347237334006997801,929721796071361437087789041,
%U A376162 79773595676787229793797978773561927,104165556509336140832819242491033872033130063,2252283824141388832759484222915451435885285752729087857
%N 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.
%C A376162 Ordered partitions are also called weak orderings.
%C A376162 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.
%C A376162 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.
%C A376162 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.
%C A376162 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
%H A376162 Andrew Howroyd, <a href="/A376162/b376162.txt">Table of n, a(n) for n = 1..20</a>
%F A376162 a(n) <= A000670(n*(n+1)/2).
%e A376162 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).
%e A376162 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.
%e A376162 From _Andrew Howroyd_, Sep 15 2024: (Start)
%e A376162 The a(3) = 3 symmetric matrices are:
%e A376162     [1 2 3]   [1 2 3]   [1 2 4]
%e A376162     [2 3 4]   [2 4 5]   [2 3 5]
%e A376162     [3 4 5]   [3 5 6]   [4 5 6]
%e A376162 (End)
%o A376162 (PARI) \\ See PARI link in A374514 for program code.
%o A376162 vector(10, n, A376162(n)) \\ _Andrew Howroyd_, Sep 16 2024
%Y A376162 Cf. A000670, A006541, A003121, A374514.
%K A376162 nonn
%O A376162 1,3
%A A376162 _Kevin O'Bryant_, Sep 12 2024
%E A376162 a(7) onwards from _Andrew Howroyd_, Sep 15 2024