# [seqfan] Stirling numbers of first kind

Joerg Arndt arndt at jjj.de
Tue Oct 6 04:00:27 CEST 2009

```In  http://www.research.att.com/~njas/sequences/A000254
the offset appears to be wrong, it should be 1

s(n,k) := numof length-n permutations into k cycles
("Stirling cycle numbers")

n:   total   m=  1      2      3     4     5     6    7   8   9
1:       1       1
2:       2       1      1
3:       6       2      3      1
4:      24       6     11      6     1
5:     120      24     50     35    10     1
6:     720     120    274    225    85    15     1
7:    5040     720   1764   1624   735   175    21    1
8:   40320    5040  13068  13132  6769  1960   322   28   1
9:  362880   40320 109584 118124 67284 22449  4536  546  36   1

EGF (serlaplace((-log(1-x))^2)/2!)
0 + 0*x +  x^2 + 3*x^3 + 11*x^4 + 50*x^5 + 274*x^6 + ...

Top comment should be
"number of permutations of n elements into 2 cycles"

For:
A000399 Stirling numbers of first kind s(n,3).
http://www.research.att.com/~njas/sequences/A000399
EGF is
(serlaplace((-log(1-x))^3)/3!)
0+0*x+0*x^2+  x^3 + 6*x^4 + 35*x^5 + 225*x^6 + ...
and not (as given)
(serlaplace((-log(1-x))^3))
6*x^3 + 36*x^4 + 210*x^5 + 1350*x^6 + ...

Top comment should be
"number of permutations of n elements into 3 cycles"

A000454 Stirling numbers of first kind s(n,4).
top comment should be
"number of permutations of n elements into 4 cycles"

Correct EGF to (-log(1-x))^4)/4!
Correct example

A000482 Stirling numbers of first kind.
http://www.research.att.com/~njas/sequences/A000482
Add to title "s(n,5)"
EGF should be (-log(1-x))^5)/5!
add (mutatis mutandis) top comment

Similarly for
http://www.research.att.com/~njas/sequences/A001233
http://www.research.att.com/~njas/sequences/A001234

```

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