Lah numbers: Difference between revisions
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Lah numbers, sometimes referred to as |
Lah numbers, sometimes referred to as ''Stirling numbers of the third kind'', are coefficients of polynomial expansions expressing rising factorials in terms of falling factorials. |
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Unsigned Lah numbers count the number of ways a set of |
Unsigned Lah numbers count the number of ways a set of '''n''' elements can be partitioned into '''k''' non-empty linearly ordered subsets. |
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Lah numbers are closely related to Stirling numbers of the first & second kinds, and may be derived from them. |
Lah numbers are closely related to Stirling numbers of the first & second kinds, and may be derived from them. |
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Lah numbers obey |
Lah numbers obey the identities and relations: |
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L(n, 0), L(0, k) = 0 |
L(n, 0), L(0, k) = 0 # for n, k > 0 |
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L(n, n) = 1 |
L(n, n) = 1 |
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L(n, 1) = n! |
L(n, 1) = n! |
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L(n, k) = ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For unsigned Lah numbers |
L(n, k) = ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For unsigned Lah numbers |
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''or'' |
''or'' |
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L(n, k) = (-1)**n * ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For signed Lah numbers |
L(n, k) = (-1)**n * ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For signed Lah numbers |
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