A341317 Array read by antidiagonals of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition).

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 8, 8, 4, 0, 0, 5, 16, 10, 16, 5, 0, 0, 6, 17, 17, 17, 17, 6, 0, 0, 7, 18, 19, 37, 19, 18, 7, 0, 0, 8, 29, 21, 38, 38, 21, 29, 8, 0, 0, 9, 30, 30, 39, 40, 39, 30, 30, 9, 0, 0, 10, 31, 32, 67, 42, 42, 67, 32, 31, 10, 0

Offset: 0

Author

N. J. A. Sloane, Feb 17 2021

Keywords

• nonn
• tabl

Comments

Consider the semigroup S consisting of the pairs (0,0) and {(i,j): i >= j >= 1}, with componentwise products. Label the elements 0 = (0,0), 1 = (1,1), 2 = (2,1), 3 = (2,2), 4 = (3,1), 5 = (3,2), 6 = (3,3), 7 = (4,1), ... Form the array A(n,k) = label of product of n-th and k-th elements, for n>=0, k>=0, and read it by antidiagonals.

Examples

			The third and fourth elements of S are (2,2) and (3,1), and their product is (6,2), which is the 17th element.
The first few rows of the multiplication table A are:
  0, [0, 0,  0,  0,  0,  0,  0,   0,   0, ...]
  1, [0, 1,  2,  3,  4,  5,  6,   7,   8, ...]
  2, [0, 2,  7,  8, 16, 17, 18,  29,  30, ...]
  3, [0, 3,  8, 10, 17, 19, 21,  30,  32, ...]
  4, [0, 4, 16, 17, 37, 38, 39,  67,  68, ...]
  5, [0, 5, 17, 19, 38, 40, 42,  68,  70, ...]
  6, [0, 6, 18, 21, 39, 42, 45,  69,  72, ...]
  7, [0, 7, 29, 30, 67, 68, 69, 121, 122, ...]
  8, [0, 8, 30, 32, 68, 70, 72, 122, 124, ...]
  ...
The first few antidiagonals are:
   0, [0]
   1, [0, 0]
   2, [0, 1,  0]
   3, [0, 2,  2,  0]
   4, [0, 3,  7,  3,  0]
   5, [0, 4,  8,  8,  4,  0]
   6, [0, 5, 16, 10, 16,  5,  0]
   7, [0, 6, 17, 17, 17, 17,  6,  0]
   8, [0, 7, 18, 19, 37, 19, 18,  7,  0]
   9, [0, 8, 29, 21, 38, 38, 21, 29,  8, 0]
  10, [0, 9, 30, 30, 39, 40, 39, 30, 30, 9, 0]
  ...
		

References

• J. M. Howie, An Introduction to Semigroup Theory, Academic Press (1976). [Background information.]

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened

Crossrefs

Cf. A341318. See A341706 for row 2.

Main diagonal gives A341736.

Programs

Maple

# Build table of elements
M:=100; ct:=0; id[0,0]:=0; x[0]:=0; y[0]:=0;
for m from 1 to M do for n from 1 to m do
ct:=ct+1; x[ct]:=m; y[ct]:=n; id[m,n]:=ct;
od: od:
# Build multiplication table:
for m from 0 to 10 do
ro:=[];
for n from 0 to m do
a1:=x[m-n]; a2:=y[m-n]; b1:=x[n]; b2:=y[n];
c1:=a1*b1; c2:=a2*b2; d:=id[c1,c2];
ro:=[op(ro),d];
od:
lprint(m,ro);
od: