A341317 - OEIS

%I #32 Feb 19 2021 10:11:11

%S 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,

%T 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,

%U 39,40,39,30,30,9,0,0,10,31,32,67,42,42,67,32,31,10,0

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

%C 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.

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

%H Alois P. Heinz, <a href="/A341317/b341317.txt">Antidiagonals n = 0..200, flattened</a>

%e The third and fourth elements of S are (2,2) and (3,1), and their product is (6,2), which is the 17th element.

%e The first few rows of the multiplication table A are:

%e   0, [0, 0,  0,  0,  0,  0,  0,   0,   0, ...]

%e   1, [0, 1,  2,  3,  4,  5,  6,   7,   8, ...]

%e   2, [0, 2,  7,  8, 16, 17, 18,  29,  30, ...]

%e   3, [0, 3,  8, 10, 17, 19, 21,  30,  32, ...]

%e   4, [0, 4, 16, 17, 37, 38, 39,  67,  68, ...]

%e   5, [0, 5, 17, 19, 38, 40, 42,  68,  70, ...]

%e   6, [0, 6, 18, 21, 39, 42, 45,  69,  72, ...]

%e   7, [0, 7, 29, 30, 67, 68, 69, 121, 122, ...]

%e   8, [0, 8, 30, 32, 68, 70, 72, 122, 124, ...]

%e   ...

%e The first few antidiagonals are:

%e    0, [0]

%e    1, [0, 0]

%e    2, [0, 1,  0]

%e    3, [0, 2,  2,  0]

%e    4, [0, 3,  7,  3,  0]

%e    5, [0, 4,  8,  8,  4,  0]

%e    6, [0, 5, 16, 10, 16,  5,  0]

%e    7, [0, 6, 17, 17, 17, 17,  6,  0]

%e    8, [0, 7, 18, 19, 37, 19, 18,  7,  0]

%e    9, [0, 8, 29, 21, 38, 38, 21, 29,  8, 0]

%e   10, [0, 9, 30, 30, 39, 40, 39, 30, 30, 9, 0]

%e   ...

%p # Build table of elements

%p M:=100; ct:=0; id[0,0]:=0; x[0]:=0; y[0]:=0;

%p for m from 1 to M do for n from 1 to m do

%p ct:=ct+1; x[ct]:=m; y[ct]:=n; id[m,n]:=ct;

%p od: od:

%p # Build multiplication table:

%p for m from 0 to 10 do

%p ro:=[];

%p for n from 0 to m do

%p a1:=x[m-n]; a2:=y[m-n]; b1:=x[n]; b2:=y[n];

%p c1:=a1*b1; c2:=a2*b2; d:=id[c1,c2];

%p ro:=[op(ro),d];

%p od:

%p lprint(m,ro);

%p od:

%Y Cf. A341318. See A341706 for row 2.

%Y Main diagonal gives A341736.

%K nonn,tabl

%O 0,8

%A _N. J. A. Sloane_, Feb 17 2021