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A341317 Array read by antidiagonals of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A329331

Adjacent sequences:  A341314 A341315 A341316 * A341318 A341319 A341320

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Feb 17 2021

STATUS

approved

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Last modified September 21 16:31 EDT 2021. Contains 347598 sequences. (Running on oeis4.)