stability of oxidation states across and down the block
A transition metal may be capable of several oxidation states, but in any real situation one of them is usually the stable one — the form you actually find sitting in a bottle or a solution. Iron loves +2 and +3; manganese in water is happiest as +2; chromium prefers +3; permanganate's +7 manganese is a fierce oxidizer just itching to drop down. Which state is stable, and how that changes as you move across a row or down a group, is what lets a chemist predict whether a compound will last or fall apart.
Two simple ideas organize most of it. First, what stabilizes a high oxidation state versus a low one. Very high states (where the metal has given up many electrons and become strongly electron-hungry) are stabilized by small, hard, electron-rich partners that can give density back — above all oxide (O2-) and fluoride (F-). That is why manganese reaches +7 only as the oxide MnO4-, and the highest fluorides exist where high oxides do. Low and even unusual low states (where the metal holds extra electrons) are stabilized by soft pi-acceptor ligands like carbon monoxide, which siphon excess electron density off the metal through back-bonding — this is how we make compounds with the metal in oxidation state zero, such as Ni(CO)4. Second, the trend across the row: early metals reach and keep high states fairly easily (titanium is happy as +4, vanadium as +5), but by the middle the highest states become powerful oxidizers (chromate, dichromate, permanganate), and in the late metals (cobalt, nickel, copper) the +2 state dominates because higher states are too hard to reach.
Down a group the story flips toward higher states: the heavier 4d and 5d congeners hold their high oxidation states far more comfortably than the first-row metal above them. Chromium(VI) is a strong oxidant, but tungsten(VI) in WO3 is placid and ordinary; manganese(VII) is explosive in the wrong company, yet rhenium(VII) oxide is stable and unremarkable. Diagrams such as the Latimer, Frost and Pourbaix plots turn all these tendencies into a quantitative picture, letting you read off which state survives at a given pH and potential — and they reveal when a state will disproportionate, splitting into a higher and a lower state at once.
Compare chromium and tungsten, both group-6 metals. Chromium(VI) as dichromate (Cr2O7 2-) is a strong oxidizer that readily drops to green chromium(III). Tungsten(VI) as WO3 is a stable, dull yellow solid with no urge to be reduced. The high state grows calmer as you go down the group — the same trend that makes the heavier metals favor higher oxidation states.
High states are stabilized by oxide and fluoride, and grow calmer down a group: aggressive Cr(VI) versus placid W(VI).
Stable does not mean inert: a thermodynamically stable oxidation state can still react quickly, and a high state can be perfectly stable until it meets something to oxidize. The diagrams predict which state should win at equilibrium, not how fast it gets there.