Concrete mathematics : a foundation for computer science

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256 SPECIAL NUMBERS
Table 256 Second-order Eulerian triangle.
3
4
5
6
7
8
1
1 0 1 _ J .f\
1 2 0 '1 i I !
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1 8 6 0 1
1 22 58 24 0 :\I '
1 52 328 444 120 0
1 114 1452 4400 3708 720 0
1 240 5610 32120 58140 33984 5040 0
1 494 19950 195800 644020 785304 341136 40320 0
We needn’t dwell further on Eulerian numbers here; it’s usually sufficient
simply to know that they exist, and to have a list of basic identities to fall
back on when the need arises. However, before we leave this topic, we should
take note of yet another triangular pattern of coefficients, shown in Table 256.
We call these “second-order Eulerian numbers” ((F)), because they satisfy a
recurrence similar to (6.35) but with n replaced by 2n - 1 in one place:
((E)) = (k+l)((n~1))+(2n-l-k)((~-:)>. (6.41)
These numbers have a curious combinatorial interpretation, first noticed by
Gessel and Stanley [118]: If we form permutations of the multiset (1, 1,2,2,
. ,n,n} with the special property that all numbers between the two occurrences
of m are greater than m, for 1 6 m 6 n, then ((t)) is the number of
such permutations that have k ascents. For example, there are eight suitable
single-ascent permutations of {l , 1,2,2,3,3}:
113322, 133221, 221331, 221133, 223311, 233211, 331122, 331221.
Thus ((T)) = 8. The multiset {l, 1,2,2,. . . , n, n} has a total of
= (2n-1)(2n-3)...(l) = y (6.42)
suitable permutations, because the two appearances of n must be adjacent
and there are 2n - 1 places to insert them within a permutation for n - 1.
For example, when n = 3 the permutation 1221 has five insertion points,
yielding 331221, 133221, 123321, 122331, and 122133. Recurrence (6.41) can
be proved by extending the argument we used for ordinary Eulerian numbers.
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