, ▁then ▁▁▁ ▁counts ▁the ▁number ▁of ▁E uler ian ▁orient ations ▁of ▁G . ▁Here ▁ ▁is ▁the ▁number ▁of ▁connected ▁components ▁of ▁G . ▁ ▁( 3 , 3 ) ▁If ▁G ▁is ▁the ▁m   ×   n ▁grid ▁graph , ▁then ▁ ▁counts ▁the ▁number ▁of ▁ways ▁to ▁tile ▁a ▁rectangle ▁of ▁width ▁ 4 m ▁and ▁height ▁ 4 n ▁with ▁T - t et rom ino es . ▁ ▁If ▁G ▁is ▁a ▁plan ar ▁graph , ▁then ▁ ▁equals ▁the ▁sum ▁over ▁weight ed ▁E uler ian ▁orient ations ▁in ▁a ▁med ial ▁graph ▁of ▁G , ▁where ▁the ▁weight ▁of ▁an ▁orientation ▁is ▁ 2 ▁to ▁the ▁number ▁of ▁s addle ▁vertices ▁of ▁the ▁orientation ▁( that ▁is , ▁the ▁number ▁of ▁vertices ▁with ▁incident ▁edges ▁cy clic ly ▁ordered ▁" in , ▁out , ▁in ▁out "). ▁ ▁Pot ts ▁and ▁Is ing ▁models ▁ ▁Define ▁the ▁hyper bol a ▁in ▁the ▁x y − plane : ▁ ▁the ▁T ut te ▁polynomial ▁special ises ▁to ▁the ▁partition ▁function , ▁ ▁of ▁the ▁Is ing ▁model ▁studied ▁in ▁statistical ▁physics . ▁Specifically , ▁along ▁the ▁hyper bol a ▁ ▁the ▁two ▁are ▁related ▁by ▁the ▁equation : ▁ ▁In ▁particular , ▁ ▁for ▁all ▁complex ▁α . ▁ ▁More ▁generally , ▁if ▁for ▁any ▁positive ▁integer ▁q , ▁we ▁define ▁the ▁hyper bol a : ▁ ▁then ▁the ▁T ut te ▁polynomial ▁special ises ▁to ▁the ▁partition ▁function ▁of ▁the ▁q - state ▁Pot