Fuss–Catalantal är inom kombinatoriken tal på formen
A
m
(
p
,
r
)
≡
r
m
p
+
r
(
m
p
+
r
m
)
=
r
m
!
∏
i
=
1
m
−
1
(
m
p
+
r
−
i
)
.
{\displaystyle A_{m}(p,r)\equiv {\frac {r}{mp+r}}{\binom {mp+r}{m}}={\frac {r}{m!}}\prod _{i=1}^{m-1}(mp+r-i).}
De är uppkallade efter N. I. Fuss och Eugène Charles Catalan .
För delindex
m
≥
0
{\displaystyle m\geq 0}
A
m
(
1
,
1
)
=
1
,
1
,
1
,
1
,
1
,
…
{\displaystyle A_{m}(1,1)=1,1,1,1,1,\ldots }
A
m
(
1
,
2
)
=
1
,
2
,
3
,
4
,
5
,
6
,
7
…
{\displaystyle A_{m}(1,2)=1,2,3,4,5,6,7\ldots }
A
m
(
2
,
1
)
=
1
,
1
,
2
,
5
,
14
,
42
,
132
,
429
,
1430
,
4862
…
{\displaystyle A_{m}(2,1)=1,1,2,5,14,42,132,429,1430,4862\ldots }
A000108
A
m
(
2
,
3
)
=
1
,
3
,
9
,
28
,
90
,
297
,
1001
,
3432
,
11934
,
41990
,
149226
,
…
{\displaystyle A_{m}(2,3)=1,3,9,28,90,297,1001,3432,11934,41990,149226,\ldots }
A000245
A
m
(
2
,
4
)
=
1
,
4
,
14
,
48
,
165
,
572
,
2002
,
7072
,
25194
,
90440
,
…
{\displaystyle A_{m}(2,4)=1,4,14,48,165,572,2002,7072,25194,90440,\ldots }
A002057
A
m
(
3
,
2
)
=
1
,
2
,
7
,
30
,
143
,
728
,
3876
,
21318
,
120175
,
690690
,
…
{\displaystyle A_{m}(3,2)=1,2,7,30,143,728,3876,21318,120175,690690,\ldots }
A006013
A
m
(
3
,
3
)
=
1
,
3
,
12
,
55
,
273
,
1428
,
7752
,
43263
,
246675
,
1430715
,
8414640
,
…
{\displaystyle A_{m}(3,3)=1,3,12,55,273,1428,7752,43263,246675,1430715,8414640,\ldots }
A001764
A
m
(
3
,
4
)
=
1
,
4
,
18
,
88
,
455
,
2448
,
13566
,
76912
,
444015
,
2601300
,
…
{\displaystyle A_{m}(3,4)=1,4,18,88,455,2448,13566,76912,444015,2601300,\ldots }
A006629
A
m
(
4
,
2
)
=
1
,
2
,
9
,
52
,
340
,
2394
,
17710
,
135720
,
1068012
,
…
{\displaystyle A_{m}(4,2)=1,2,9,52,340,2394,17710,135720,1068012,\ldots }
A069271
A
m
(
p
,
r
)
=
A
m
(
p
,
r
−
1
)
+
A
m
−
1
(
p
,
p
+
r
−
1
)
{\displaystyle A_{m}(p,r)=A_{m}(p,r-1)+A_{m-1}(p,p+r-1)}
börjar med
A
−
1
(
p
,
r
)
=
0
{\displaystyle A_{-1}(p,r)=0}
A
m
(
p
,
p
)
=
A
m
+
1
(
p
,
1
)
{\displaystyle A_{m}(p,p)=A_{m+1}(p,1)}
Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia , Fuss–Catalan number , 29 december 2013 . Fuss, N. I. (1791). ”Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat”. Nova Acta Academiae Sci. Petropolitanae 9: sid. 243–251. Bisch, Dietmar; Jones, Vaughan (1997). ”Algebras associated to intermediate subfactors”. Invent. Mathem. 128 (1): sid. 89–157. doi :10.1007/s002220050137 Przytycki, Jozef H.; Sikora, Adam S. (2000). ”Polygon dissections and Euler, Fuss, Kirkman , and Cayley Numbers”. J. Combinat. Theory A 92: sid. 68–76. doi :10.1006/jcta.1999.3042 Aval, Jean-Christophe (2008). ”Multivariate Fuss-Catalan numbers”. Discr. Math. 20 (308): sid. 4660–4669. doi :10.1016/j.disc.2007.08.100 Alexeev, N.; Götze, F; Tikhomirov, A. (2010). ”Asymptotic distribution of singular values of powers of random matrices”. Lith. Math. J. 50 (2): sid. 121–132. doi :10.1007/s10986-010-9074-4 Mlotkowski, Wojciech (2010). ”Fuss-Catalan Numbers in noncommutative probability” . Docum. Mathm. 15: sid. 939–955. https://eudml.org/doc/222801 . Gordon, Ian G.; Griffeth, Stephen (2012). ”Catalan numbers for complex reflection groups”. Am. J. Math. 134 (6): sid. 1491–1502. doi :10.1353/ajm.2012.0047 