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In combinatorial mathematics, the Lobb number ''L''''m'',''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. 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 ''n''th Catalan number equals the Lobb number ''L''0,''n''. They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the ''n''th Catalan number. The Lobb numbers are parameterized by two non-negative integers ''m'' and ''n'' with ''n'' ≥ ''m'' ≥ 0. The (''m'', ''n'')th Lobb number ''L''''m'',''n'' is given in terms of binomial coefficients by the formula : 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== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lobb numbers」の詳細全文を読む スポンサード リンク
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