# bettiStats -- statistics on Betti tables of a sample of monomial ideals or list of objects

## Synopsis

• Usage:
bettiStats(List)
• Inputs:
• Optional inputs:
• CountPure => ..., default value false, optional input to show the number of objects in the list whose Betti tables are pure
• IncludeZeroIdeals => ..., default value true, optional input to choose whether zero ideals should be included
• SaveBettis => ..., default value , optional input to store all Betti tables computed
• Verbose => ..., default value false, optional input to request verbose feedback
• Outputs:
• , of objects of type BettiTally, representing the mean Betti table shape and the mean Betti table of the elements in the list L.

## Description

For a sample of ideals stored as a list, this function computes some basic Betti table statistics of the sample. Namely, it computes the average shape of the Betti tables (where 1 is recorded in entry (ij) for each element if $beta_{ij}$ is not zero), and it also computes the average Betti table (that is, the table whose (ij) entry is the mean value of $beta_{ij}$ for all ideals in the sample).

 i1 : R = ZZ/101[a..e]; i2 : L={monomialIdeal"a2b,bc", monomialIdeal"ab,bc3",monomialIdeal"ab,ac,bd,be,ae,cd,ce,a3,b3,c3,d3,e3"} 2 3 3 o2 = {monomialIdeal (a b, b*c), monomialIdeal (a*b, b*c ), monomialIdeal (a , ------------------------------------------------------------------------ 3 3 3 3 a*b, b , a*c, c , b*d, c*d, d , a*e, b*e, c*e, e )} o2 : List i3 : (meanBettiShape,meanBetti,stdDevBetti) = bettiStats L; i4 : meanBettiShape 0 1 2 3 4 5 o4 = total: 1 2 2 1.33333 1.33333 .333333 0: 1 . . . . . 1: . 1 .333333 .333333 .333333 . 2: . .666667 .666667 .333333 .333333 . 3: . .333333 .666667 .333333 .333333 . 4: . . .333333 .333333 .333333 .333333 o4 : BettiTally i5 : meanBetti 0 1 2 3 4 5 o5 = total: 1 5.33333 10.3333 10 5 1 0: 1 . . . . . 1: . 3 3.66667 2 .333333 . 2: . 2 5 4.33333 1.33333 . 3: . .333333 .666667 .666667 .333333 . 4: . . 1 3 3 1 o5 : BettiTally i6 : stdDevBetti 1 2 3 4 5 o6 = total: 5.46008 13.4481 14.1421 7.07107 1.41421 1: 2.82843 5.18545 2.82843 .471405 . 2: 2.16025 6.37704 6.12826 1.88562 . 3: .471405 .471405 .942809 .471405 . 4: . 1.41421 4.24264 4.24264 1.41421 o6 : BettiTally

For sample size $N$, the average Betti table shape considers nonzero Betti numbers. It is to be interpreted as follows: entry (i,j) encodes the following sum of indicators: $\sum_{all ideals} 1_{beta_{ij}>0} / N$; that is, the proportion of ideals with a nonzero $beta_{ij}$. Thus an entry of 0.33 means 33% of ideals have a non-zero Betti number there.

 i7 : apply(L,i->betti res i) 0 1 2 0 1 2 0 1 2 3 4 5 o7 = {total: 1 2 1, total: 1 2 1, total: 1 12 29 30 15 3} 0: 1 . . 0: 1 . . 0: 1 . . . . . 1: . 1 . 1: . 1 . 1: . 7 11 6 1 . 2: . 1 1 2: . . . 2: . 5 14 13 4 . 3: . 1 1 3: . . 1 2 1 . 4: . . 3 9 9 3 o7 : List i8 : meanBettiShape 0 1 2 3 4 5 o8 = total: 1 2 2 1.33333 1.33333 .333333 0: 1 . . . . . 1: . 1 .333333 .333333 .333333 . 2: . .666667 .666667 .333333 .333333 . 3: . .333333 .666667 .333333 .333333 . 4: . . .333333 .333333 .333333 .333333 o8 : BettiTally

For sample size $N$, the average Betti table is to be interpreted as follows: entry $(i,j)$ encodes $\sum_{I\in ideals}beta_{ij}(R/I) / N$:

 i9 : apply(L,i->betti res i) 0 1 2 0 1 2 0 1 2 3 4 5 o9 = {total: 1 2 1, total: 1 2 1, total: 1 12 29 30 15 3} 0: 1 . . 0: 1 . . 0: 1 . . . . . 1: . 1 . 1: . 1 . 1: . 7 11 6 1 . 2: . 1 1 2: . . . 2: . 5 14 13 4 . 3: . 1 1 3: . . 1 2 1 . 4: . . 3 9 9 3 o9 : List i10 : meanBetti 0 1 2 3 4 5 o10 = total: 1 5.33333 10.3333 10 5 1 0: 1 . . . . . 1: . 3 3.66667 2 .333333 . 2: . 2 5 4.33333 1.33333 . 3: . .333333 .666667 .666667 .333333 . 4: . . 1 3 3 1 o10 : BettiTally

## Ways to use bettiStats :

• "bettiStats(List)"

## For the programmer

The object bettiStats is .