This document is one of More SageMath Tutorials. You may edit it on github. $$\def\NN{\mathbb{N}}$$ $$\def\ZZ{\mathbb{Z}}$$ $$\def\QQ{\mathbb{Q}}$$ $$\def\RR{\mathbb{R}}$$ $$\def\CC{\mathbb{C}}$$

# Counting¶

sage: Partitions(100000).cardinality()


# Random generation¶

sage: M = BinaryTrees(100)
sage: t = M.random_element()
sage: view(t, tightpage=True)


# Species¶

sage: from sage.combinat.species.library import *
sage: BT = CombinatorialSpecies()
sage: Leaf =  SingletonSpecies()
sage: BT.define(Leaf+(BT*BT))

sage: o = var("o")
sage: BT.isotypes([o]*5).list()
[o*(o*(o*(o*o))), o*(o*((o*o)*o)), o*((o*o)*(o*o)), o*((o*(o*o))*o), o*(((o*o)*o)*o), (o*o)*(o*(o*o)), (o*o)*((o*o)*o), (o*(o*o))*(o*o), ((o*o)*o)*(o*o), (o*(o*(o*o)))*o, (o*((o*o)*o))*o, ((o*o)*(o*o))*o, ((o*(o*o))*o)*o, (((o*o)*o)*o)*o]

sage: S = BT.isotype_generating_series()
sage: S
x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + O(x^6)

sage: S[100]
227508830794229349661819540395688853956041682601541047340

sage: S
x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + ... + O(x^101)


# Words¶

sage: m = WordMorphism('a->acabb,b->bcacacbb,c->baba')
sage: m.fixed_point('a')
word: acabbbabaacabbbcacacbbbcacacbbbcacacbbac...


# Lattice points of polytopes¶

sage: A = random_matrix(ZZ, nrows=6,ncols=3,x=7)
sage: L = LatticePolytope(A)
sage: L.plot3d()

sage: L.npoints()  # should be cardinality!
28


This example used PALP and J-mol

# Graphs up to an isomorphism¶

sage: show(graphs(5, lambda G: G.size() <= 4))


# Guessing and the like¶

sage: data = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

sage: oeis(data)
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: from ore_algebra import guess, OreAlgebra
sage: guess(data, OreAlgebra(ZZ['x'], 'Dx'))
(x^2 + x - 1)*Dx + 2*x + 1

sage: L = _

sage: x = var('x')
sage: F = function('F')(x)
sage: equation = sum( L[i] * diff(F, x, i) for i in range(L.order()+1) )
sage: equation
(2*x + 1)*F(x) + (x^2 + x - 1)*D[0](F)(x)

sage: solution = desolve(equation, F, ics=[0,1])
sage: solution
-1/(x^2 + x - 1)

sage: solution.series(x)
1 + 1*x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 + 34*x^8 + 55*x^9 + 89*x^10 + 144*x^11 + 233*x^12 + 377*x^13 + 610*x^14 + 987*x^15 + 1597*x^16 + 2584*x^17 + 4181*x^18 + 6765*x^19 + Order(x^20)

sage: L.rational_solutions()
[(1/(x^2 + x - 1),)]

sage: L.generalized_series_solutions(15)
[1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 13*x^6 + 21*x^7 + 34*x^8 + 55*x^9 + 89*x^10 + 144*x^11 + 233*x^12 + 377*x^13 + 610*x^14 + O(x^15)]

sage: L.generalized_series_solutions(1000)
[1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + ... + O(x^1000)]

sage: L = guess(data, OreAlgebra(ZZ['n'], 'Sn'))
sage: L
-Sn^2 + Sn + 1

sage: L.generalized_series_solutions()
[(1.618033988749895?)^n*(1 + O(n^(-5))), (-0.618033988749895?)^n*(1 + O(n^(-5)))]

sage: s = L.generalized_series_solutions()[0]
sage: s.exponential_part()
1.618033988749895?

sage: s.exponential_part().minpoly()
x^2 - x - 1

sage: data = [1, 1, 2, 5, 14]

sage: oeis(data)
0: A000108: Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
1: A120588: G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, with a(0) = 1.
2: A080937: Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

sage: data = _[0].first_terms()
sage: data
(1, 1, 2, 5, 14, 42, 132, ...)

sage: L = guess(data, OreAlgebra(ZZ['x'], 'Dx'))
sage: L
(4*x^2 - x)*Dx^2 + (10*x - 2)*Dx + 2

sage: L.power_series_solutions(10)
[1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + O(x^9)]
sage: L.power_series_solutions(1000)
[1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ... + O(x^1000)]

sage: equation = sum( L[i] * diff(F, x, i) for i in range(L.order()+1) ); equation
2*(5*x - 1)*D[0](F)(x) + (4*x^2 - x)*D[0, 0](F)(x) + 2*F(x)

sage: res = desolve(equation, F)
sage: res

sage: res = res.subs(_K2=I/2, _K1=-1)
sage: res.series(x)
1 + 1*x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ... + 1767263190*x^19 + Order(x^20)

1/2*(I*sqrt(4*x - 1) + 1)/x

sage: M = guess(data, OreAlgebra(ZZ['n'], 'Sn'))
sage: M
(-n - 2)*Sn + 4*n + 2


Asymptotic expansion:

sage: M.generalized_series_solutions()
[4^n*n^(-3/2)*(1 - 9/8*n^(-1) + 145/128*n^(-2) - 1155/1024*n^(-3) + 36939/32768*n^(-4) + O(n^(-5)))]

sage: M.generalized_series_solutions(10)
[4^n*n^(-3/2)*(1 - 9/8*n^(-1) + 145/128*n^(-2) - 1155/1024*n^(-3) + 36939/32768*n^(-4) - 295911/262144*n^(-5) + 4735445/4194304*n^(-6) - 37844235/33554432*n^(-7) + 2421696563/2147483648*n^(-8) - 19402289907/17179869184*n^(-9) + O(n^(-10)))]


# Symmetric functions¶

sage: S = SymmetricFunctions(QQ)
sage: S.inject_shorthands()


Arithmetic:

sage: s[2,1] * s[2,1]
s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]

sage: p( m[1] * ( e[3]*s[2] + 1 ))
p[1] + 1/12*p[1, 1, 1, 1, 1, 1] - 1/6*p[2, 1, 1, 1, 1] - 1/4*p[2, 2, 1, 1] + 1/6*p[3, 1, 1, 1] + 1/6*p[3, 2, 1]


Hopf structure:

sage: s[3,2,1].coproduct()
s[] # s[3, 2, 1] + s[1] # s[2, 2, 1] + s[1] # s[3, 1, 1] + s[1] # s[3, 2] + s[1, 1] # s[2, 1, 1] + s[1, 1] # s[2, 2] + s[1, 1] # s[3, 1] + s[1, 1, 1] # s[2, 1] + s[2] # s[2, 1, 1] + s[2] # s[2, 2] + s[2] # s[3, 1] + s[2, 1] # s[1, 1, 1] + 2*s[2, 1] # s[2, 1] + s[2, 1] # s[3] + s[2, 1, 1] # s[1, 1] + s[2, 1, 1] # s[2] + s[2, 2] # s[1, 1] + s[2, 2] # s[2] + s[2, 2, 1] # s[1] + s[3] # s[2, 1] + s[3, 1] # s[1, 1] + s[3, 1] # s[2] + s[3, 1, 1] # s[1] + s[3, 2] # s[1] + s[3, 2, 1] # s[]


Plethysms:

sage: p[3](p[6])
sage: s[3,2](s[2,1]+p[3])