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26 Combinatorial AnalysisNotation

§26.1 Special Notation

(For other notation see Notation for the Special Functions.)

x real variable.
h,j,k,,m,n nonnegative integers.
λ integer partition.
π plane partition.
|A| number of elements of a finite set A.
j|k j divides k.
(h,k) greatest common divisor of positive integers h and k.

The main functions treated in this chapter are:

(mn) binomial coefficient.
(mn1,n2,,nk) multinomial coefficient.
mn Eulerian number.
[mn]q Gaussian polynomial.
B(n) Bell number.
C(n) Catalan number.
p(n) number of partitions of n.
pk(n) number of partitions of n into at most k parts.
pp(n) number of plane partitions of n.
s(n,k) Stirling numbers of the first kind.
S(n,k) Stirling numbers of the second kind.

Alternative Notations

Many combinatorics references use the rising and falling factorials:

26.1.1 xn¯ =x(x+1)(x+2)(x+n-1),
xn¯ =x(x-1)(x-2)(x-n+1).

Other notations for s(n,k), the Stirling numbers of the first kind, include Sn(k) (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), Snk (Jordan (1939), Moser and Wyman (1958a)), (n-1k-1)Bn-k(n) (Milne-Thomson (1933)), (-1)n-kS1(n-1,n-k) (Carlitz (1960), Gould (1960)), (-1)n-k[nk] (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).

Other notations for S(n,k), the Stirling numbers of the second kind, include 𝒮n(k) (Fort (1948)), 𝔖nk (Jordan (1939)), σnk (Moser and Wyman (1958b)), (nk)Bn-k(-k) (Milne-Thomson (1933)), S2(k,n-k) (Carlitz (1960), Gould (1960)), {nk} (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).