|
| |
|
|
A036039
|
|
Triangle of multinomial coefficients read by rows.
|
|
39
| |
|
|
1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,4
|
|
|
COMMENTS
| The sequence of row lengths is A000041(n), n>=1, (partition numbers).
Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 12 2006
A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
|
|
|
REFERENCES
| Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".
|
|
|
LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Tom Copeland, Lagrange a la Lah
Mark Dominus Cycle classes of permutations [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jun 26 2009]
W. Lang, First ten rows and polynomials.
|
|
|
FORMULA
| Raising and lowering operators are given for the partition polynomials formed from A036039 in the link in "Lagrange a la Lah Part I" on pg. 23. - Tom Copeland, Sep 18 2011
|
|
|
EXAMPLE
| 1; 1,1; 2,3,1; 6,8,3,6,1; 24,30,20,20,15,10,1; ...
|
|
|
MATHEMATICA
| Mathematica from Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jun 26 2009, Jun 27 2009 (Start)
aspartitions[n_]:=Reverse/@Sort[Sort/@Partitions[n]]; (* Abramowitz & Stegun ordering *);
ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!
Range[n]^#)&/@Function[par, Count[par, # ]&/@Range[n]]/@aspartitions[n])
The function "ascycleclasses" is then identical with A&S multinomial M2. (End)
|
|
|
CROSSREFS
| Cf. A036036-A036040.
Cf. A102189 (rows reversed).
Sequence in context: A076631 A035485 A074306 * A092271 A054115 A100822
Adjacent sequences: A036036 A036037 A036038 * A036040 A036041 A036042
|
|
|
KEYWORD
| nonn,easy,nice,tabf
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net).
|
| |
|
|
