Conditions state we assume the # of rows is in $a, so not counting the initial assignment, 41 characters:

$a=5;
print map{@_=$j+1..($j+=++$i);"@_\n"}1..$a

#Update 2 chars better (39)
print map{"@{[$j+1..($j+=++$i)]}\n"}1..$a
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Update: blakem pointed out that mine doesn't align the numbers. Darn.
Another update: I don't see any reason why any one of the following does not work, there seems to be a bug with the deprecated '$#' variable. In fact, the one with 's' segfaults on 5.6.1 (though if it worked, it would left justify the numbers). It woulda been 64 characters (if someone submits a patch to the perl core, they can have the credit for this :)

$#="%@{[2*length $a]}d";print map"@{[$j+1..($j+=++$i)]}\n",1..$a
$#="%@{[2*length $a]}u";print map"@{[$j+1..($j+=++$i)]}\n",1..$a
$#="%@{[2*length $a]}s";print map"@{[$j+1..($j+=++$i)]}\n",1..$a
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Last Update: Thanks to help from master caddy Albannach (for figuring out that 'f' works), 60 characters:

$#=" %@{[2*length$a]}.f";print map{$j+1..($j+=++$i),$/}1..$a
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Really last update: Following the letter AND spirit of that last 'updated' condition, at 67 characters:

$#=" %@{[length int$a*$a/2]}.f";print map{$j+1..($j+=++$i),$/}1..$a
# or
$#=" %@{[length$a++*$a/2]}.f";print map{$j+1..($j+=++$i),$/}1..$a-1
# or
$#=" %@{[length$a*($a+1)/2]}.f";print map{$j+1..($j+=++$i),$/}1..$a
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Tired of updating :) But more caddy help, this time from blakem(65 characters):

$#=" %@{[length$a++*$a/2]}g";map{print$j+1..($j+=++$i),$/}1..$a-1
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