Layered permutations and rational generating functions Anders Björner Department of Mathematics Kungliga Tekniska Högskolan S-100 44 Stockholm, SWEDEN and Bruce Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 www.math.msu.edu/ ˜ sagan August 28, 2006
Outline
Let P be the positive integers.
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N .
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N .
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 .
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 . � 2 N − 1 Theorem if N ≥ 1 c N = if N = 0 . 1
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 . � 2 N − 1 Theorem if N ≥ 1 c N = if N = 0 . 1 So we have the rational generating function c N x N = 1 − x � 1 − 2 x . N ≥ 0
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 . � 2 N − 1 Theorem if N ≥ 1 c N = if N = 0 . 1 So we have the rational generating function c N x N = 1 − x � 1 − 2 x . N ≥ 0 Questions: 1. Is this an isolated incident or part of a larger picture?
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 . � 2 N − 1 Theorem if N ≥ 1 c N = if N = 0 . 1 So we have the rational generating function c N x N = 1 − x � 1 − 2 x . N ≥ 0 Questions: 1. Is this an isolated incident or part of a larger picture? 2. What does this have to do with patterns in permutations?
Let P be the positive integers. A composition of a non-negative integer N is a sequence w = k 1 k 2 . . . k r with all k i ∈ P and � i k i = N . Let c N be the number of compositions of N . Ex. If N = 3 then c 3 = 4 counting compositions 3 , 21 , 12 , 111 . � 2 N − 1 Theorem if N ≥ 1 c N = if N = 0 . 1 So we have the rational generating function c N x N = 1 − x � 1 − 2 x . N ≥ 0 Questions: 1. Is this an isolated incident or part of a larger picture? 2. What does this have to do with patterns in permutations? Moral: It can be better to count by containment instead of avoidance.
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] .
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] . Call π ∈ S n layered if it has the form π = p , p − 1 , . . . , 1 , p + q , p + q − 1 , . . . , p + 1 , p + q + r , . . . for certain p , q , r , . . . called the layer lengths .
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] . Call π ∈ S n layered if it has the form π = p , p − 1 , . . . , 1 , p + q , p + q − 1 , . . . , p + 1 , p + q + r , . . . for certain p , q , r , . . . called the layer lengths . There is a bijection between layered permutations and compositions by π ← → w = pqr . . .
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] . Call π ∈ S n layered if it has the form π = p , p − 1 , . . . , 1 , p + q , p + q − 1 , . . . , p + 1 , p + q + r , . . . for certain p , q , r , . . . called the layer lengths . There is a bijection between layered permutations and compositions by π ← → w = pqr . . . Ex. π = 3 2 1 5 4 9 8 7 6 ← → w = 3 2 4.
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] . Call π ∈ S n layered if it has the form π = p , p − 1 , . . . , 1 , p + q , p + q − 1 , . . . , p + 1 , p + q + r , . . . for certain p , q , r , . . . called the layer lengths . There is a bijection between layered permutations and compositions by π ← → w = pqr . . . Ex. π = 3 2 1 5 4 9 8 7 6 ← → w = 3 2 4. Any set A (the alphabet) has Kleene closure A ∗ = { w = k 1 k 2 . . . k r | k i ∈ A for all i and r ≥ 0 } .
Let [ n ] = { 1 , 2 , . . . , n } and let S n be the symmetric group on [ n ] . Call π ∈ S n layered if it has the form π = p , p − 1 , . . . , 1 , p + q , p + q − 1 , . . . , p + 1 , p + q + r , . . . for certain p , q , r , . . . called the layer lengths . There is a bijection between layered permutations and compositions by π ← → w = pqr . . . Ex. π = 3 2 1 5 4 9 8 7 6 ← → w = 3 2 4. Any set A (the alphabet) has Kleene closure A ∗ = { w = k 1 k 2 . . . k r | k i ∈ A for all i and r ≥ 0 } . Note w is a composition iff w ∈ P ∗ .
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset).
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995):
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r .
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w .
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w . Ex. If u = 4 1 3 and w = 4 1 4 3 2 4 2 then u ≤ w ,
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w . Ex. If u = 4 1 3 and w = 4 1 4 3 2 4 2 then u ≤ w , for example, 1 2 3 4 5 6 7 w = 4 1 4 3 2 4 2
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w . Ex. If u = 4 1 3 and w = 4 1 4 3 2 4 2 then u ≤ w , for example, 1 2 3 4 5 6 7 w = 4 1 4 3 2 4 2 ≥ ≥ ≥ u = 4 1 3
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w . Ex. If u = 4 1 3 and w = 4 1 4 3 2 4 2 then u ≤ w , for example, 1 2 3 4 5 6 7 w = 4 1 4 3 2 4 2 I = { 3 , 5 , 6 } . and ≥ ≥ ≥ u = 4 1 3
Letting π ≤ σ whenever π is a pattern in σ turns S = ⊎ n ≥ 0 S n into a partially ordered set (poset). This induces a partial order on P ∗ (Bergeron, Bousquet-Mélou, and Dulucq, 1995): If u = k 1 . . . k r and w = l 1 . . . l s then u ≤ w iff there is a subsequence l i 1 . . . l i r of w with k j ≤ l i j for 1 ≤ j ≤ r . The index set I = { i 1 , . . . , i r } is called an embedding of u into w . Ex. If u = 4 1 3 and w = 4 1 4 3 2 4 2 then u ≤ w , for example, 1 2 3 4 5 6 7 w = 4 1 4 3 2 4 2 I = { 3 , 5 , 6 } . and ≥ ≥ ≥ u = 4 1 3 Given u ≤ w there is a unique rightmost embedding, I, such that I ≥ I ′ componentwise for all embeddings I ′ . The embedding above is rightmost.
P ∗ = ǫ
P ∗ = 1 ǫ
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