A341736 -id:A341736 - cp's OEIS frontend

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

N. J. A. Sloane, Feb 17 2021

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.

			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]
  ...

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

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

Cf. A341318. See A341706 for row 2.

Main diagonal gives A341736.

# 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:

A341318 Lower triangular table of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 7, 0, 3, 8, 10, 0, 4, 16, 17, 37, 0, 5, 17, 19, 38, 40, 0, 6, 18, 21, 39, 42, 45, 0, 7, 29, 30, 67, 68, 69, 121, 0, 8, 30, 32, 68, 70, 72, 122, 124, 0, 9, 31, 34, 69, 72, 75, 123, 126, 129, 0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136, 0, 11, 46, 47, 106, 107, 108, 191, 192, 193, 194, 301

Offset: 0

N. J. A. Sloane, Feb 17 2021

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), ... The triangle gives T(n,k) = label of product of n-th and k-th elements, for n>=k>=0.

See A341317 for further information, including a Maple program.

			Triangle begins:
0, [0]
1, [0, 1]
2, [0, 2, 7]
3, [0, 3, 8, 10]
4, [0, 4, 16, 17, 37]
5, [0, 5, 17, 19, 38, 40]
6, [0, 6, 18, 21, 39, 42, 45]
7, [0, 7, 29, 30, 67, 68, 69, 121]
8, [0, 8, 30, 32, 68, 70, 72, 122, 124]
9, [0, 9, 31, 34, 69, 72, 75, 123, 126, 129]
10, [0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136]
...

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

Cf. A341317, A341706.

Main diagonal gives A341736.

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)][])(f(n)):
T:= (n, k)-> (h-> t(h[1]*h[3])+h[2]*h[4])(map(g, [n, k])):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 17 2021

t[n_] := n*(n - 1)/2;
f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
g[n_] := Function[x, {x, n - t[x]}][f[n]];
T[n_, k_] := (Function[h, t[h[[1]]*h[[3]]] + h[[2]]*h[[4]]])[Flatten @ Map[g, {n, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

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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).

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