Lobb number
Lobb number
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In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.[1]
Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L0,n.[2] They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the nth Catalan number.[3]
The Lobb numbers are parameterized by two non-negative integers m and n with n ≥ m ≥ 0. The (m, n)th Lobb number Lm,n is given in terms of binomial coefficients by the formula
- Lm,n=2m+1m+n+1(2nm+n) for n≥m≥0.{displaystyle L_{m,n}={frac {2m+1}{m+n+1}}{binom {2n}{m+n}}qquad {text{ for }}ngeq mgeq 0.}
As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which n + m copies of the value +1 and n − m copies of the value −1 may be arranged into a sequence such that all of the partial sums of the sequence are non-negative.
References[edit]
^ Koshy, Thomas (March 2009). "Lobb's generalization of Catalan's parenthesization problem". The College Mathematics Journal. 40 (2): 99–107. doi:10.4169/193113409X469532..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Koshy, Thomas (2008). Catalan Numbers with Applications. Oxford University Press. ISBN 978-0-19-533454-8.
^ Lobb, Andrew (March 1999). "Deriving the nth Catalan number". Mathematical Gazette. 83 (8): 109–110.
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