It's been well documented that the Philadelphia Flyers have been generally trending upwards since an extended period of poor play in late October and early November. The reasons for the team's improvement are numerous - improved scoring, the return of key players like Sean Couturier, the emergence of Shayne Gostisbehere.

For their part, the players have long attributed the bounce in play to increased familiarity with the systems of first-year NHL head coach Dave Hakstol. Direction from Hakstol was becoming habit, and the team was able to execute effectively without thinking, resulting in quicker breakouts, cleaner passes, and better offensive zone play.

One obvious disadvantage to hiring a head coach without any NHL coaching experience was the fact that he was coming in with little preexisting knowledge of his opponents. Sure, Hakstol would have assistant coaches to help him, but there would likely be a learning curve as the new coach became more familiar with the other NHL teams, their players, and the preferred systems of opposing coaches.

Now past the halfway point of the season, the Flyers have reached the point of their schedule where they will regularly face teams for the second, third and even fourth times under the direction of Dave Hakstol. But will this prove advantageous for Philadelphia, giving Hakstol a chance to better understand and prepare for his opponents? Or could it actually be a negative, as other teams are now more familiar with Hakstol and his tactics?

Let's take a look at the numbers so far.

Flyers have played much better in second, third games against opponents

In 2015-16 thus far, the Philadelphia Flyers have faced 16 teams more than once. They've then played those sixteen teams a combined twenty more times this season. The win-loss records show that Philadelphia has achieved much better standings results the second (and third) time they've faced an opponent versus matchup No. 1.

Matchup Type Record First Game 5-9-2 Rematches 12-5-3

That's a point percentage of 37.5% the first time the Flyers face a team, and a 67.5% rate in rematches. Per Hockey Reference, the former rate would place Philadelphia dead-last in the overall standings, while the latter would rank the Flyers third-best in the league.

Still, we're dealing with very small samples here. The outcomes could be driven by factors that are very difficult to attribute to coaching, such as fantastic (or terrible) goaltending performances, or unsustainable shooting percentages. To that end, let's take a look at one of the most sustainable measures of team-level play - score-adjusted shot attempt differential at five-on-five (per War-On-Ice).

Matchup Type Score-Adjusted Corsi First Game 47.59% Rematches 51.77%

The Flyers have legitimately improved their underlying puck possession metrics in repeat matchups against opponents. It's not just Steve Mason and Michal Neuvirth, or uncharacteristically strong special teams play, or even an elevated shooting percentage. Philadelphia's play at even strength under Hakstol has been much better the second time facing a team.

Real effect, or just noise?

There is undoubtedly a disparity between how the Flyers have played in their first matchup against an opponent versus the second and third times that they've faced teams this season. But that disparity does not prove that increased familiarity is the primary reason for the improvement. And with the sample sizes so small, it would be impossible to prove mathematically that this is a driving force behind the Flyers' improvement.

Still, the narrative does make some sense. After all, the idea that a sharp coach would benefit from having first-hand observation and tape study of opponents does not seem farfetched.

Hakstol himself was skeptical of any connection, but he did note that the disparity was striking.

"Honestly I don’t [think it's the reason]. Not at all. I didn’t know that stat - that’s an interesting stat. Worthy of taking a look at, but I don’t think so."

Veteran forward Ryan White, on the other hand, wondered if there may be something to the theory.

"I think for sure [Hakstol] is learning - he’s hasn’t seen a lot of these teams before. I think he’s done a real good job of gameplanning and putting in the right system every night."

"At the same time, you see a team once, and then you see them a second time, you know their tendencies. Like I said, he’s been doing a good job gameplanning for the teams when we play them a second time. I don’t know if it’s a coincidence or what. Again, any time you see [a team] the first time, it’s kind of a fresh game. After that, you try to pick up their tendencies and go try to limit what they want to do. I don’t know if it’s a stat or a secret or what, but it seems to be working, I guess."

White also proposed an alternate theory that falls more in line with the general consensus from the locker room over the past few months.

"We’ve got our base system that we’ve been playing all the time, and I think probably the biggest thing is that we’ve gotten better at the system throughout the season. I think it's the main thing. Relying on it a lot more, and knowing the ins and outs of it, when to step up, when not to, it’s made the game a little bit easier."

That's certainly a possibility. By the nature of the schedule, most of Philadelphia's "first game" matchups occurred early in the year, when the team as a whole was struggling. Maybe the team's better performance in rematches is less a result of Hakstol's tape study and gameplanning, and more due to improved performance across the board as the season has progressed.

If the theory holds water, however, it would bode well for the remainder of the Flyers' season. Only a late February game against the Arizona Coyotes would be a "first-time" matchup for Hakstol and Philadelphia. If the coach is truly gaining an advantage or "catching up" to the rest of the league with additional experience, the next three months could be quite exciting for Flyers fans.