Description
A Betti tally is a special type of
Tally that is printed as a display of graded Betti numbers. The class was created so the function
betti could return something that both prints nicely and from which information can be extracted. The keys are triples
(i,d,h), where
i is the homological degree,
d is a list of integers giving a multidegree, and
h is the result of applying a weight covector to
d. Only
i and
h are used in printing.
i1 : t = new BettiTally from { (0,{0},0) => 1, (1,{1},1) => 2, (2,{3},3) => 3, (2,{4},4) => 4 }
0 1 2
o1 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o1 : BettiTally

i2 : peek oo
o2 = BettiTally{(0, {0}, 0) => 1}
(1, {1}, 1) => 2
(2, {3}, 3) => 3
(2, {4}, 4) => 4

For convenience, the operations of direct sum (
++), tensor product (
**),
codim,
degree,
dual,
hilbertPolynomial,
hilbertSeries,
pdim,
poincare,
regularity, and degree shifting (numbers in brackets or parentheses), have been implemented for Betti tallies. These operations mimic the corresponding operations on chain complexes.
i3 : t(5)
0 1 2
o3 = total: 1 2 7
5: 1 2 .
4: . . 3
3: . . 4
o3 : BettiTally

i4 : t[5]
5 6 7
o4 = total: 1 2 7
5: 1 2 .
4: . . 3
3: . . 4
o4 : BettiTally

i5 : t ++ oo
0 1 2 3 4 5 6 7
o5 = total: 1 2 7 . . 1 2 7
5: . . . . . 1 2 .
4: . . . . . . . 3
3: . . . . . . . 4
2: . . . . . . . .
1: . . . . . . . .
0: 1 2 . . . . . .
1: . . 3 . . . . .
2: . . 4 . . . . .
o5 : BettiTally

i6 : t ** t
0 1 2 3 4
o6 = total: 1 4 18 28 49
0: 1 4 4 . .
1: . . 6 12 .
2: . . 8 16 9
3: . . . . 24
4: . . . . 16
o6 : BettiTally

i7 : dual t
2 1 0
o7 = total: 7 2 1
2: 4 . .
1: 3 . .
0: . 2 1
o7 : BettiTally

i8 : regularity t
o8 = 2

A Betti tally can be multiplied by an integer or by a rational number, and the values can be lifted to integers, when possible.
i9 : (1/2) * t
0 1 2
o9 = total: 1/2 1 7/2
0: 1/2 1 .
1: . . 3/2
2: . . 2
o9 : BettiTally

i10 : 2 * oo
0 1 2
o10 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o10 : BettiTally

i11 : lift(oo,ZZ)
0 1 2
o11 = total: 1 2 7
0: 1 2 .
1: . . 3
2: . . 4
o11 : BettiTally

Various combinations of the degree vectors can be displayed by using
betti(BettiTally).