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

0,8

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.

References

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

Links

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

Example

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]
  ...

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:

Crossrefs

Cf. A341318. See A341706 for row 2.

Main diagonal gives A341736.

Sequence in context: A265080 A228275 A228250 * A101164 A229079 A357144

Adjacent sequences: A341314 A341315 A341316 * A341318 A341319 A341320

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 17 2021

Status

approved