The information that you are asking about is what sort of inferences can be made about the character of Colonel Sartoris based on the information that the story presents via indirect and direct characterization. If this is correct, let's first see the circumstances in which the character appears in the story, and then part from there.
The character of Colonel Sartoris appears in parts I and IV of "A Rose for Emily". In part I, it is established that he was the mayor of Jefferson as of 1894.
Judging by the fact that he prefers to use his military rank to identify himself (he is "Colonel Sartoris" before, during, and after he was Mayor) the first inference about his character is that he is very likely to have served in the Civil War as part of the Confederate Army. He also holds on to this rank as part of his persona as a way to honor his service, and to keep alive the memory of the defunct Confederacy. This indirectly would describe Sartoris as the epitomical “Southern gentleman”, which is a motif that is always present in Faulkner’s work.
More seasoned Faulkner readers know that this is actually true, as John Sartoris is one of Faulkner's most enigmatic characters featured in an eponymous novel, Sartoris, which further expands upon the man's military career.
Another inference that can be made is that Sartoris had ties with the Griersons' that go back, perhaps, to the War days, or his military service. These ties must have been strong enough that Sartoris decided that, upon Emily's father's death, she would no longer need to pay taxes, claiming that the city is indebted to the man (the father).
Colonel Sartoris invented an involved tale to the effect that Miss Emily's father had loaned money to the town, which the town, as a matter of business, preferred this way of repaying. Only a man of Colonel Sartoris' generation and thought could have invented it, and only a woman could have believed it.
The people accepted the ruling, Colonel Sartoris felt the need to make it, and it all leads us to believe that there may have been a debt to be paid, indeed, but it perhaps did not involve any money given to the city. Could it be a promise made to Grierson upon his death? Could it have been an act of guilt over something done to the Griersons?
Back to the story, it can also be inferred that Colonel Sartoris’s ties to the Confederacy have kept his ideals about racial separatism very much alive. He is, after all, the same man who
fathered the edict that no Negro woman should appear on the streets without an apron
It is clear that blacks had no rights during that time, but they were no longer slaves. For him to impose a ruling that involves humiliation and labeling, not to mention that the ruling makes no sense at all, shows that there is a part of him that is precisely that: nonsensical and archaic.
In part IV, we find that the connection between Sartoris's contemporaries and the Griersons is tight enough for their daughters and granddaughters to be sent to Emily's house for china-painting lessons. The inference that can be made from this is that the people who were from Sartoris’s generation felt a sense of camaraderie, having all being part of a time period that is disappearing in front of their eyes. Perhaps this is a legacy left by Sartoris: not to leave one of “their own” behind, and to support them in any way they can.
She fitted up a studio in one of the downstairs rooms, where the daughters and granddaughters of Colonel Sartoris' contemporaries were sent to her with the same regularity and in the same spirit that they were sent to church on Sundays with a twenty-five-cent piece for the collection plate. Meanwhile her taxes had been remitted.
Even ten years after Sartoris had been dead, Emily would still hold tight to his ruling of not taxing her.
On and all, the most accurate inference that can be concluded regarding the character of Colonel Sartoris is that he is a very influential figure in Jefferson, whose dispensations toward the Griersons may be also symbols of traditional, conservative Southern loyalty. So influential was his support that the people still remember it. They even end up humanizing Emily and, perhaps, it was thanks to his support to the Griersons that others followed suit.
Since we do not know the real reason why he made those dispensations, let us just assume that he did them in good spirit to help someone whom he knew would eventually fall on dire straits. Perhaps he already knew that the Griersons would end with Emily, alone, poor, and with a potential to go crazy like her aunt once did.