[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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