Catalan Numbers Catalan Numbers Murray Bremner University of Saskatchewan, Canada University of Cape Town, Monday 30 November 2015 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc Catalan Numbers PART 1 Catalan numbers arise from a simple enumeration problem: In how many different ways can we compute the product of n factors which don't commute? With three factors abc there are two multiplications to perform and we can do them either in one order (ab)c or the other a(bc). We can represent these possibilities as rooted planar binary trees: ∗ ∗ ∗ c a ∗ ab bc The asterisks represent multiplications, and the two subtrees below each asterisk represent the two factors being multiplied. ∗ ∗ ∗ ∗ d ∗ d ∗ ∗ ∗ c a ∗ ab cd ab bc ∗ ∗ a ∗ a ∗ ∗ d b ∗ bc cd Catalan Numbers With four factors, there are five ways to compute the product: ∗ ∗ a ∗ a ∗ ∗ d b ∗ bc cd Catalan Numbers With four factors, there are five ways to compute the product: ∗ ∗ ∗ ∗ d ∗ d ∗ ∗ ∗ c a ∗ ab cd ab bc Catalan Numbers With four factors, there are five ways to compute the product: ∗ ∗ ∗ ∗ d ∗ d ∗ ∗ ∗ c a ∗ ab cd ab bc ∗ ∗ a ∗ a ∗ ∗ d b ∗ bc cd ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? First, let's consider some other applications of Catalan numbers. Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? Catalan Numbers Using parentheses these sequences of multiplications look like this: ((ab)c)d; (a(bc))d; (ab)(cd); a((bc)d); a(b(cd)): For 5, 6, 7, 8, 9, 10, . factors the number of possibilities is 14; 42; 132; 429; 1430; 4862;::: Is there a simple compact formula for the terms of this sequence? If there is, how do we go about finding it? First, let's consider some other applications of Catalan numbers. This follows from the formula we will prove and Stirling's formula: p nn n! ∼ 2πn as n ! 1 e The Catalan numbers appear as the solution to a very large number of different combinatorial problems. Richard Stanley's Enumerative Combinatorics: Volume 2 (Cambridge U. Press, 1999) has an exercise which gives 66 different interpretations of the Catalan numbers. The Catalan numbers appear as sequence A000108 in the OEIS (On-Line Encyclopedia of Integer Sequences, oeis.org): \This is probably the longest entry in the OEIS, and rightly so." Catalan Numbers Asymptotics: 4n C ∼ p as n ! 1 n n3=2 π The Catalan numbers appear as the solution to a very large number of different combinatorial problems. Richard Stanley's Enumerative Combinatorics: Volume 2 (Cambridge U. Press, 1999) has an exercise which gives 66 different interpretations of the Catalan numbers. The Catalan numbers appear as sequence A000108 in the OEIS (On-Line Encyclopedia of Integer Sequences, oeis.org): \This is probably the longest entry in the OEIS, and rightly so." Catalan Numbers Asymptotics: 4n C ∼ p as n ! 1 n n3=2 π This follows from the formula we will prove and Stirling's formula: p nn n! ∼ 2πn as n ! 1 e Richard Stanley's Enumerative Combinatorics: Volume 2 (Cambridge U. Press, 1999) has an exercise which gives 66 different interpretations of the Catalan numbers. The Catalan numbers appear as sequence A000108 in the OEIS (On-Line Encyclopedia of Integer Sequences, oeis.org): \This is probably the longest entry in the OEIS, and rightly so." Catalan Numbers Asymptotics: 4n C ∼ p as n ! 1 n n3=2 π This follows from the formula we will prove and Stirling's formula: p nn n! ∼ 2πn as n ! 1 e The Catalan numbers appear as the solution to a very large number of different combinatorial problems. The Catalan numbers appear as sequence A000108 in the OEIS (On-Line Encyclopedia of Integer Sequences, oeis.org): \This is probably the longest entry in the OEIS, and rightly so." Catalan Numbers Asymptotics: 4n C ∼ p as n ! 1 n n3=2 π This follows from the formula we will prove and Stirling's formula: p nn n! ∼ 2πn as n ! 1 e The Catalan numbers appear as the solution to a very large number of different combinatorial problems.