Eulerian Numbers versus Stirling Numbers of the First Kind
Eulerian Numbers versus Stirling Numbers of the First Kind
In a permutation , an ascent is a pair with <. The Eulerian number counts the number of permutations of with exactly ascents, . Alternatively, counts the number of permutations of with exactly ' permutation runs, . For example, the permutation has the three ascents , , and and the two runs and .
(,,…,)
a
1,
a
2
a
n
(,)
a
m
a
m+1
a
m
a
m+1
n |
k |
{1,2,3,…,n}
k
0≤k≤n-1
n |
k'-1 |
{1,2,3,…,n}
k
1≤k'≤n
(1,2,3,5,4)
(1,2)
(2,3)
(3,5)
(1,2,3,5)
(4)
The unsigned Stirling number of the first kind counts the number of permutations of whose cycle decomposition has cycles. For example, the permutation is the mapping , , , , , so its cycle decomposition is with four cycles.
n |
k |
{1,2,3,…,n}
k
(1,2,3,5,4)
11
22
33
45
54
(1)(2)(3)(45)
Both types of numbers count the permutations of in different ways. Define the number to count the number of permutations that have cycles and runs. This Demonstration lays out these numbers in a square; the row sums are and the column sums are ; the sums of either of those is .
n!
{1,2,3,…,n}
n |
i,j |
i
j
n |
i |
n |
j |
n!