* algebra/irsn.spad.pamphlet (IrrRepSymNatPackage): Tidy.

	* algebra/partperm.spad.pamphlet (PartitionsAndPermutations): Likewise.
	* algebra/cycles.spad.pamphlet (complete$CycleIndicators): Now
	take only positive integers.
	(powerSum$CycleIndicators): Likewise.
	(elementary$CycleIndicators): Likewise.
	(alternating$CycleIndicators): Likewise.
	(cyclic$CycleIndicators): Likewise.
	(dihedral$CycleIndicators): Likewise.
	(graphs$CycleIndicators): Likewise.

1 files changed, 28 insertions, 27 deletions

diff --git a/src/algebra/cycles.spad.pamphlet b/src/algebra/cycles.spad.pamphlet
index b8bf2e16..c2941dd3 100644
--- a/src/algebra/cycles.spad.pamphlet
+++ b/src/algebra/cycles.spad.pamphlet
@@ -33,35 +33,36 @@ CycleIndicators: Exports == Implementation where
   RN   ==> Fraction Integer
   FR   ==> Factored Integer
   macro NNI == NonNegativeInteger
+  macro PI == PositiveInteger
   Exports ==> with
  
-    complete: NNI -> SPOL RN
+    complete: PI -> SPOL RN
       ++\spad{complete n} is the \spad{n} th complete homogeneous
       ++ symmetric function expressed in terms of power sums.
       ++ Alternatively it is the cycle index of the symmetric
       ++ group of degree n.
  
-    powerSum: NNI -> SPOL RN
+    powerSum: PI -> SPOL RN
       ++\spad{powerSum n} is the \spad{n} th power sum symmetric
       ++ function.
  
-    elementary: NNI -> SPOL RN
+    elementary: PI -> SPOL RN
       ++\spad{elementary n} is the \spad{n} th elementary symmetric
       ++ function expressed in terms of power sums.
  
-    alternating: NNI -> SPOL RN
+    alternating: PI -> SPOL RN
       ++\spad{alternating n} is the cycle index of the
       ++ alternating group of degree n.
  
-    cyclic: NNI -> SPOL RN    --cyclic group
+    cyclic: PI -> SPOL RN    --cyclic group
       ++\spad{cyclic n} is the cycle index of the
       ++ cyclic group of degree n.
  
-    dihedral: NNI -> SPOL RN    --dihedral group
+    dihedral: PI -> SPOL RN    --dihedral group
       ++\spad{dihedral n} is the cycle index of the
       ++ dihedral group of degree n.
  
-    graphs: NNI -> SPOL RN
+    graphs: PI -> SPOL RN
       ++\spad{graphs n} is the cycle index of the group induced on
       ++ the edges of a graph by applying the symmetric function to the
       ++ n nodes.
@@ -83,7 +84,7 @@ CycleIndicators: Exports == Implementation where
       ++ of the two groups whose cycle indices are \spad{s1} and
       ++ \spad{s2}.
  
-    SFunction:L I -> SPOL RN
+    SFunction:L PI -> SPOL RN
       ++\spad{SFunction(li)} is the S-function of the partition \spad{li}
       ++ expressed in terms of power sum symmetric functions.
  
@@ -99,24 +100,24 @@ CycleIndicators: Exports == Implementation where
     trm: PTN -> SPOL RN
     trm pt == monomial(inv(pdct(pt) :: RN),pt)
  
-    list: Stream L I -> L L I
+    list: Stream L PI -> L L PI
     list st == entries complete st
  
     complete i ==
       i=0 => 1
-      +/[trm(partition pt) for pt in list(partitions i)]
+      +/[trm partition pt for pt in list partitions i]
  
  
-    even?: L I -> B
+    even?: L PI -> B
     even? li == even?( #([i for i in li | even? i]))
  
     alternating i ==
-      2 * _+/[trm(partition li) for li in list(partitions i) | even? li]
+      2 * _+/[trm partition li for li in list partitions i | even? li]
 
     elementary i ==
        i=0 => 1
-       +/[(spol := trm(partition pt); even? pt => spol; -spol)
-                          for pt in list(partitions i)]
+       +/[(spol := trm partition pt; even? pt => spol; -spol)
+                          for pt in list partitions i]
  
     divisors: I -> L I
     divisors n ==
@@ -125,23 +126,23 @@ CycleIndicators: Exports == Implementation where
                  [[a.factor**j for j in 1..a.exponent] for a in b]);
       if #(b) = 1 then c else concat(n,c)
  
-    ss: (I,I) -> SPOL RN
+    ss: (PI,I) -> SPOL RN
     ss(n,m) ==
-      li : L I := [n for j in 1..m]
+      li : L PI := [n for j in 1..m]
       monomial(1,partition li)
  
     powerSum n == ss(n,1)
  
     cyclic n ==
       n = 1 => powerSum 1
-      +/[(eulerPhi(i) / n) * ss(i,numer(n/i)) for i in divisors n]
+      +/[(eulerPhi(i) / n) * ss(i::PI,numer(n/i)) for i in divisors n]
  
     dihedral n ==
       k := n quo 2
       odd? n => (1/2) * cyclic n + (1/2) * ss(2,k) * powerSum 1
       (1/2) * cyclic n + (1/4) * ss(2,k) + (1/4) * ss(2,k-1) * ss(1,2)
  
-    trm2: L I -> SPOL RN
+    trm2: L PI -> SPOL RN
     trm2 li ==
       lli := powers(partition li)$PTN
       xx := 1/(pdct partition li)
@@ -151,12 +152,12 @@ CycleIndicators: Exports == Implementation where
         k := ll0 quo 2
         c :=
           odd? ll0 => ss(ll0,ll1 * k)
-          ss(k,ll1) * ss(ll0,ll1 * (k - 1))
+          ss(k::PI,ll1) * ss(ll0,ll1 * (k - 1))
         c := c * ss(ll0,ll0 * ((ll1*(ll1 - 1)) quo 2))
         prod2 : SPOL RN := 1
         for r in lli | first(r) < ll0 repeat
           r0 := first r; r1 := second r
-          prod2 := ss(lcm(r0,ll0),gcd(r0,ll0) * r1 * ll1) * prod2
+          prod2 := ss(lcm(r0,ll0)::PI,gcd(r0,ll0) * r1 * ll1) * prod2
         prod := c * prod2 * prod
       xx * prod
  
@@ -180,24 +181,24 @@ CycleIndicators: Exports == Implementation where
  
     cap(spol1,spol2) == eval cup(spol1,spol2)
  
-    mtpol: (I,SPOL RN) -> SPOL RN
+    mtpol: (PI,SPOL RN) -> SPOL RN
     mtpol(n,spol)==
       zero? spol => 0
-      deg := partition [n*k for k in (degree spol)::L(I)]
+      deg := partition [n*k for k in (degree spol)::L(PI)]
       monomial(leadingCoefficient spol,deg) + mtpol(n,reductum spol)
  
-    evspol: ((I -> SPOL RN),SPOL RN) -> SPOL RN
+    evspol: ((PI -> SPOL RN),SPOL RN) -> SPOL RN
     evspol(fn2,spol) ==
       zero? spol => 0
       lc := leadingCoefficient spol
-      prod := */[fn2 i for i in (degree spol)::L(I)]
+      prod := */[fn2 i for i in (degree spol)::L(PI)]
       lc * prod + evspol(fn2,reductum spol)
  
     wreath(spol1,spol2) == evspol(mtpol(#1,spol2),spol1)
  
     SFunction li==
       a:Matrix SPOL RN :=
-        matrix [[complete((k -j+i)::NNI) for k in li for j in 1..#li]
+        matrix [[complete((k -j+i)::PI) for k in li for j in 1..#li]
                     for i in 1..#li]
       determinant a
  
@@ -210,7 +211,7 @@ CycleIndicators: Exports == Implementation where
       #li1 < #li2 =>
         error "skewSFunction: partition1 does not include partition2"
       li2:=roundup (li1,li2)
-      a:Matrix SPOL RN:=matrix [[complete((k-li2.i-j+i)::NNI)
+      a:Matrix SPOL RN:=matrix [[complete((k-li2.i-j+i)::PI)
                for k in li1 for j in 1..#li1]  for i in 1..#li1]
       determinant a
 
@@ -252,7 +253,7 @@ EvaluateCycleIndicators(F):T==C where
        pt:PTN
        spol:SPOL RN
        i:I
-       evp(fn, pt)== */[fn i for i in pt::(L I)]
+       evp(fn, pt)== */[fn i for i in pt::L(PositiveInteger)]
  
        eval(fn,spol)==
         if spol=0