Martino inherited a team from the ailing Tito Vilanova that secured a fourth La Liga title in five years last season but had suffered a humiliating 7-0 drubbing at the hands of eventual winners Bayern Munich in the two-legged semi-final of Europe's elite club competition.
The 3-0 home defeat in the second leg ended a 21-match unbeaten run at their giant arena and the aggregate result was by far their worst in Europe.
"They were just the better team in all respects, physically and in football terms too," Barca's World Player of the Year Lionel Messi said on UEFA's official website last week.
"They just played a great first leg and... were really superior to us," added the Argentine, who like compatriot Martino hails from the city of Rosario.
The frailty at the back and inability to put away chances that characterised some of Barca's performances last season has lingered into this term and Martino will want to start putting that right in the first meeting between the Spanish club and the Dutch side, both former continental champions.
Barca have begun their La Liga title defence with four wins in four matches but had to rely on a 94th-minute winner from substitute Alexis Sanchez to secure three points at home to Sevilla on Saturday.
They let slip a two-goal lead and some slapstick defending and poor marking meant they were seconds away from dropping their first points of the campaign.
After arriving from Newell's Old Boys, Martino said he planned to stick with Barca's attacking style of football based on quick passing and long periods of possession.
However, he has also instructed his players to close down opponents with the same bite they showed under Pep Guardiola that was perhaps missing at times last season under Vilanova.
"Sevilla's goals were a result of a lack of control," Martino said after the match.
"We won the game but we need to work on some situations," he added after Sevilla defender Coke was left unmarked at a corner and stole in to make it 2-2 late in the game. "We had some problems in the rearguard."
There were some positives to take away from the game, particularly a fine performance from new signing Neymar, who set up Messi for Barca's second goal and looked dangerous throughout.
The Brazil forward brought the fans to their feet with one sublime piece of skill when he flicked a team mate's pass between his legs and into space before speeding off and leaving his hapless marker floundering.
"Neymar played an impeccable match because it's not just about his attacking game but the peripheral work he does for the team," Martino said.
Ajax coach Frank de Boer, who is bidding to lead the Dutch champions into the knockout round for the first time since 2005-06, will be a familiar face to many of the Barca fans.
The 43-year-old former Netherlands international played for Barca between 1999 and 2003 and won the Spanish league title in 1999 alongside twin brother Ronald.
Siem de Jong is available just weeks after being hospitalised with a collapsed lung.
The influential midfielder made a near miraculous recovery capped by an appearance as a substitute in Saturday's 2-1 win against PEC Zwolle.
De Boer said if Ajax were to achieve anything in Barcelona they would have to be much more convincing.
"If Messi has seen these images of us, I doubt he'll be too concerned," he said. "We had a lot of good luck."
One Dutchman who almost certainly will not be at Wednesday's game is former Ajax, Barcelona and Netherlands great Johan Cruyff, who was Barca coach when they won the first of four European crowns in 1992.
Cruyff has fallen out with Barca President Sandro Rosell and was quoted as saying in the Spanish media last week he will not set foot in the Nou Camp while Rosell remains in charge.
Italian club AC Milan and Scottish side Celtic are the other two teams in Group H.
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