September Morn Update
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Ingy döt Net
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6856 changed files with 141342 additions and 21127 deletions
1
Task/Paraffins/00META.yaml
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1
Task/Paraffins/00META.yaml
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--- {}
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42
Task/Paraffins/Julia/paraffins.julia
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42
Task/Paraffins/Julia/paraffins.julia
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const branches = 4
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const nmax = 500
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const rooted = zeros(BigInt, nmax + 1)
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const unrooted = zeros(BigInt, nmax + 1)
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rooted[1] = rooted[2] = unrooted[1] = unrooted[2] = 1
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const c = zeros(BigInt, branches)
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function tree(br, n, l, sum, cnt)
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for b in br+1:branches
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sum += n
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if (sum > nmax) || (l * 2 >= sum && b >= branches)
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return
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elseif b == br + 1
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c[br + 1] = rooted[n + 1] * cnt
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else
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c[br + 1] *= rooted[n + 1] + b - br - 1
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c[br + 1] = div(c[br + 1], b - br)
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end
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if l*2 < sum
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unrooted[sum + 1] += c[br + 1]
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end
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if b < branches
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rooted[sum + 1] += c[br + 1]
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end
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for m in n-1:-1:1
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tree(b, m, l, sum, c[br + 1])
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end
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end
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end
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bicenter(n) = if iseven(n) unrooted[n + 1] += div(rooted[div(n, 2) + 1] * (rooted[div(n, 2) + 1] + 1), 2) end
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function paraffins()
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for n in 1:nmax
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tree(0, n, n, 1, one(BigInt))
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bicenter(n)
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println("$n: $(unrooted[n + 1])")
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end
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end
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paraffins()
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@ -1,14 +1,10 @@
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s[m_, p_, n_] :=
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CycleIndexPolynomial[SymmetricGroup[m],
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Table[ComposeSeries[p, x^i + O[x]^(n + 1)], {i, m}]];
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G000598[n_] := Nest[1 + x s[3, #, n] &, 1 + O[x], n];
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G000602[n_] :=
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Block[{x},
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x*CycleIndexPolynomial[SymmetricGroup[4],
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Table[ComposeSeries[#, x^i + O[x]^(n + 1)], {i, 4}]] -
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CycleIndexPolynomial[SymmetricGroup[2],
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Table[ComposeSeries[# - 1, x^i + O[x]^(n + 1)], {i, 2}]] +
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ComposeSeries[#, x^2 + O[x]^(n + 1)] &@
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Fold[Series[
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1 + x/6 (#1^3 + 3 #1 ComposeSeries[#1, x^2 + O[x]^#2] +
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2 ComposeSeries[#1, x^3 + O[x]^#2]), {x, 0, #2}] &,
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1 + O[x], Range[n + 1]]];
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x s[4, #, n] - s[2, # - 1, n] +
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ComposeSeries[#, x^2 + O[x]^(n + 1)] &[G000598[n]];
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A000602[n_] := SeriesCoefficient[G000602[n], n];
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A000602List[n_] := CoefficientList[G000602[n], x];
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Grid@Transpose@{Range[0, 200], A000602List@200}
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46
Task/Paraffins/Phix/paraffins.phix
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Task/Paraffins/Phix/paraffins.phix
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constant MAX_N = 32,
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BRANCH = 4
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sequence {rooted,unrooted} @= repeat(0,MAX_N+2)
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procedure tree(integer br, n, l=n, tot=1, atom cnt=1)
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atom c
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for b=br+1 to BRANCH do
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tot += n
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if tot>=MAX_N+1
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or (l*2>=tot and b>=BRANCH) then
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return
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end if
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if b==br+1 then
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c = rooted[n+1]*cnt
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else
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c *= (rooted[n+1]+(b-br-1))/(b-br)
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end if
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if l*2<tot then
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unrooted[tot+1] += c
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end if
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if b<BRANCH then
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rooted[tot+1] += c
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for m=1 to n-1 do
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tree(b,m,l,tot,c)
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end for
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end if
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end for
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end procedure
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procedure bicenter(integer s)
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if mod(s,2)=0 then
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atom aux = rooted[s/2+1]
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unrooted[s+1] += aux*(aux+1)/2
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end if
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end procedure
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rooted[1..2] = 1
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unrooted[1..2] = 1
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for n=1 to MAX_N do
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tree(0, n)
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bicenter(n)
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if n<10 or n=MAX_N then
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printf(1,"%d: %d\n",{n, unrooted[n+1]})
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end if
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end for
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32
Task/Paraffins/Python/paraffins-3.py
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Task/Paraffins/Python/paraffins-3.py
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#!/usr/bin/python3
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from functools import lru_cache
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def Z_S(n, f, k):
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"""
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The cycle index of the symmetric group has recurrence
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Z(S_n, f(x)) = 1/n \sum_{i=1}^n f(x^i) Z(S_{n-i}, f(x)).
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This function finds the coefficient of x^k in Z(S_n, f(x))
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"""
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# Special case to avoid division by zero
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if n == 0:
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return 1 if k == 0 else 0
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# Special case as a speed optimisation
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if n == 1:
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return f(k)
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return sum(
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sum(f(ij // i) * Z_S(n-i, f, k - ij) for ij in range(0, k+1, i))
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for i in range(1, n+1)
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) // n
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@lru_cache(maxsize=None)
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def A000598(k): return 1 if k == 0 else Z_S(3, A000598, k-1)
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@lru_cache(maxsize=None)
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def A000642(k): return Z_S(2, A000598, k)
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def A000631(k): return Z_S(2, A000642, k)
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def A000602(k): return A000642(k) + (A000642((k-1) // 2) if k % 2 == 1 else 0) - A000631(k-1)
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for k in range(500): print(k, A000602(k))
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@ -1,27 +1,28 @@
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/*REXX program enumerates (without repetition) the # of paraffins with N atoms of carbon*/
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parse arg nodes . /*obtain optional argument from the CL.*/
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if nodes=='' | nodes=="," then nodes=100 /*Not specified? Then use the default.*/
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rooted. = 0; rooted.0=1; rooted.1=1 /*define the base rooted numbers.*/
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unrooted. = 0; unrooted.0=1; unrooted.1=1 /* " " " unrooted " */
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numeric digits max(9,nodes%2) /*this program may use gihugeic numbers*/
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w=length(nodes) /*W: used for aligning formatted nodes.*/
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say right(0,w) unrooted.0 /*show enumerations of 0 carbon atoms*/
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if nodes=='' | nodes=="," then nodes= 100 /*Not specified? Then use the default.*/
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rooted. = 0; rooted.0= 1; rooted.1= 1 /*define the base rooted numbers.*/
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unrooted. = 0; unrooted.0= 1; unrooted.1= 1 /* " " " unrooted " */
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numeric digits max(9, nodes % 2) /*this program may use gihugeic numbers*/
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w= length(nodes) /*W: used for aligning formatted nodes*/
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say right(0, w) unrooted.0 /*show enumerations of 0 carbon atoms*/
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/* [↓] process all nodes (up to NODES)*/
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do C=1 for nodes; h=C%2 /*C: is the number of carbon atoms. */
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do C=1 for nodes; h= C % 2 /*C: is the number of carbon atoms. */
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call tree 0, C, C, 1, 1 /* [↓] if # of carbon atoms is even···*/
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if C//2==0 then unrooted.C=unrooted.C + rooted.h * (rooted.h + 1) % 2
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say right(C,w) unrooted.C /*display an aligned formatted number. */
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if C//2==0 then unrooted.C= unrooted.C + rooted.h * (rooted.h + 1) % 2
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say right(C, w) unrooted.C /*display an aligned formatted number. */
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end /*C*/
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exit /*stick a fork in it, we're all done. */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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tree: procedure expose rooted. unrooted. nodes #. /*this function is recursive.*/
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parse arg br,n,L,sum,cnt; nm=n-1; LL=L+L; brp=br+1
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do b=brp to 4; sum=sum+n; if sum>nodes then leave
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if b==4 then if LL>=sum then leave
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if b==brp then #.br=rooted.n * cnt
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else #.br=#.br * (rooted.n + b - brp) % (b - br)
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if LL<sum then unrooted.sum=unrooted.sum + #.br
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parse arg br,n,L,sum,cnt; nm= n - 1; LL= L + L; brp= br + 1
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do b=brp to 4; sum= sum + n; if sum>nodes then leave
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if b==4 then if LL>=sum then leave
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if b==brp then #.br= rooted.n * cnt
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else #.br= #.br * (rooted.n + b - brp) % (b - br)
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if LL<sum then unrooted.sum= unrooted.sum + #.br
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if b==4 then leave
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rooted.sum = rooted.sum + #.br
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do m=nm by -1 for nm; call tree b, m, L, sum, #.br; end /*m*/
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do m=nm by -1 for nm; call tree b, m, L, sum, #.br; end /*m*/
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end /*b*/ /* ↑↑↑↑↑↑↑↑↑ recursive invocation of TREE. */
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return
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