read PERCY:
print(` Support for Proofs in Foata and Zeilberger's article`):
print(`Babson and Steingrimsson's Newly Discovered ... `):
 
maj:=a_cb+b_ca+c_ba+ba: 
S11:=a_cb+2*b_ca+ba:
S13:=a_cb+2*b_ac+ab:
des:=ba:
rise:=ab:
print(`In Markovian Statistics language, maj is`):
gumaj:=BSE(maj,j,i,n):
print(gumaj):
print(`In Markovian Statistics language, des is`):
gudes:=BSE(des,j,i,n):
print(gudes):
print(`In Markovian Statistics language, rise is`):
gurise:=BSE(rise,j,i,n):
print(gurise):
#Proof of Theorem 1(the second part of Conj. 8 of [BS])
print(`Stuff for Proof of Theorem 1(the second part of Conj. 8 of [BS])`):
print(`In Markovian Statistics langauge, S11 is`):
gu11:=BSE(S11,j,i,n):
print(gu11):
lu11:=PermUmbra(gu11,q,z):
print(` Its  Umbral Operator  is F(z)->`):
print(YafeUm(lu11,F,D)):
print(`When you apply this operator to`, z*(z^(n-1)-q^(n-1))/(z-q)):
print(`You get`):
factor(expand(
simplify(ApplyUmbra(lu11, z*(z^(n-1)-q^(n-1))/(z-q),[z]),symbolic)
));
 
#Proof of Theorem 2(the third part of Conj. 8 of [BS])
print(`Stuff for Proof of Theorem 2(the third part of Conj. 8 of [BS])`):
 
print(`In Markovian Statistics langauge, S13 is`):
gu11:=BSE(S13,j,i,n):
print(gu13):
print(` Its  Umbral Operator  is`):
lu13:=PermUmbra(gu13,q,z):
print(` Its  Umbral Operator  is F(z)->`):
print(YafeUm(lu13,F,D)):
print(`When you apply this operator to`, z*(1-z^(n-1)*q^(n-1))/(1-z*q)):
print(`You get`):
factor(expand(
simplify(ApplyUmbra(lu13,z*(1-q^(n-1)*z^(n-1))/(1-q*z),[z]),symbolic)
));
 
#End of Proof of Theorem 2
 
##Theorem 3 of Foata-Zeilberger (Conj. 10 of  [BS])
print(`Stuff for the proof of Theorem 3 of Foata-Zeilberger `):
print(`The Umbral operator linking A_{n-1} to A_n, for [S5,rise] is F(z)->`):
YafeUm(PermUmbraMS([gu5,gurise],[q,t],z),F,D);
print(`The Umbral operator linking A_{n-1} to A_n, for [maj,des] is F(z)->`):
YafeUm(PermUmbraMS([gu8,gumaj],[q,t],z),F,D);
 
#end of Theorem 3
print(` The whole thing took`, time(), `seconds of CPU time`):
#quit: