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1: 26.17 The Twelvefold Way
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
labeled labeled k n ( k - n + 1 ) n k ! S ( n , k )
unlabeled labeled ( k + n - 1 n ) ( k n ) ( n - 1 n - k )
labeled unlabeled S ( n , 1 ) + S ( n , 2 ) + + S ( n , k ) { 1 n k 0 n > k S ( n , k )
unlabeled unlabeled p k ( n ) { 1 n k 0 n > k p k ( n ) - p k - 1 ( n )
2: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with N + 1 regular singularities at z = a j , j = 1 , 2 , , N , and at , is given by
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z - a j ) d w d z + ( j = 1 N q j z - a j ) w = 0 , j = 1 N q j = 0 .
The exponents at the finite singularities a j are { 0 , 1 - γ j } and those at are { α , β } , where …With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N - 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z - a j ) - γ j / 2 ) W ( z ) ,
3: 26.5 Lattice Paths: Catalan Numbers
26.5.1 C ( n ) = 1 n + 1 ( 2 n n ) = 1 2 n + 1 ( 2 n + 1 n ) = ( 2 n n ) - ( 2 n n - 1 ) = ( 2 n - 1 n ) - ( 2 n - 1 n + 1 ) .
26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n - k ) ,
26.5.4 C ( n + 1 ) = 2 ( 2 n + 1 ) n + 2 C ( n ) ,
26.5.5 C ( n + 1 ) = k = 0 n / 2 ( n 2 k ) 2 n - 2 k C ( k ) .
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
4: 31.15 Stieltjes Polynomials
If t k is a zero of the Van Vleck polynomial V ( z ) , corresponding to an n th degree Stieltjes polynomial S ( z ) , and z 1 , z 2 , , z n - 1 are the zeros of S ( z ) (the derivative of S ( z ) ), then t k is either a zero of S ( z ) or a solution of the equation … The zeros z k , k = 1 , 2 , , n , of the Stieltjes polynomial S ( z ) are the critical points of the function G , that is, points at which G / ζ k = 0 , k = 1 , 2 , , n , where … then there are exactly ( n + N - 2 N - 2 ) polynomials S ( z ) , each of which corresponds to each of the ( n + N - 2 N - 2 ) ways of distributing its n zeros among N - 1 intervals ( a j , a j + 1 ) , j = 1 , 2 , , N - 1 . … If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index m = ( m 1 , m 2 , , m N - 1 ) , where each m j is a nonnegative integer, there is a unique Stieltjes polynomial with m j zeros in the open interval ( a j , a j + 1 ) for each j = 1 , 2 , , N - 1 . … Let S m ( z ) and S l ( z ) be Stieltjes polynomials corresponding to two distinct multi-indices m = ( m 1 , m 2 , , m N - 1 ) and l = ( 1 , 2 , , N - 1 ) . …
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6: 26.8 Set Partitions: Stirling Numbers
s ( n , k ) denotes the Stirling number of the first kind: ( - 1 ) n - k times the number of permutations of { 1 , 2 , , n } with exactly k cycles. … where ( x ) n is the Pochhammer symbol: x ( x + 1 ) ( x + n - 1 ) . … For n 1 , … uniformly for n = o ( k 1 / 2 ) . For asymptotic approximations for s ( n + 1 , k + 1 ) and S ( n , k ) that apply uniformly for 1 k n as n see Temme (1993) and Temme (2015, Chapter 34). …
7: 24.5 Recurrence Relations
24.5.1 k = 0 n - 1 ( n k ) B k ( x ) = n x n - 1 , n = 2 , 3 , ,
24.5.5 k = 0 n ( n k ) 2 k E n - k + E n = 2 .
24.5.6 k = 2 n ( n k - 2 ) B k k = 1 ( n + 1 ) ( n + 2 ) - B n + 1 , n = 2 , 3 , ,
24.5.7 k = 0 n ( n k ) B k n + 2 - k = B n + 1 n + 1 , n = 1 , 2 , ,
24.5.8 k = 0 n 2 2 k B 2 k ( 2 k ) ! ( 2 n + 1 - 2 k ) ! = 1 ( 2 n ) ! , n = 1 , 2 , .
8: 26.6 Other Lattice Path Numbers
D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
26.6.10 D ( m , n ) = D ( m , n - 1 ) + D ( m - 1 , n ) + D ( m - 1 , n - 1 ) , m , n 1 ,
9: 24.15 Related Sequences of Numbers
24.15.7 B n = k = 0 n ( - 1 ) k ( n + 1 k + 1 ) S ( n + k , k ) / ( n + k k ) ,
24.15.8 k = 0 n ( - 1 ) n + k s ( n + 1 , k + 1 ) B k = n ! n + 1 .
24.15.10 2 n - 1 4 n p 2 B 2 n S ( p + 2 n , p - 1 ) ( mod p 3 ) , 2 2 n p - 3 .
The Fibonacci numbers are defined by u 0 = 0 , u 1 = 1 , and u n + 1 = u n + u n - 1 , n 1 . The Lucas numbers are defined by v 0 = 2 , v 1 = 1 , and v n + 1 = v n + v n - 1 , n 1 . …
10: 26.11 Integer Partitions: Compositions
For example, there are eight compositions of 4: 4 , 3 + 1 , 1 + 3 , 2 + 2 , 2 + 1 + 1 , 1 + 2 + 1 , 1 + 1 + 2 , and 1 + 1 + 1 + 1 . … c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . …
F 1 = 1 ,
F n = F n - 1 + F n - 2 , n 2 .