Two Binomial Coefficient Analogues Bruce Sagan Department of - PowerPoint PPT Presentation

(c) Lattice paths. A NE lattice path of length n is a squence P = s 1 . . . s n starting at (0 , 0) and with each s i being a unit step north ( N ) or east ( E ). Ex. We have P = ENEEN = Let P n , k = { P = s 1 . . . s n : P has k N -steps and n − k E -steps } There is a bijection f : P n , k → W n , k where w = f ( P ) is obtained by replacing each N by a 0 and each E by a 1. Ex. We have f 1 1 0 P = ENEEN ← → w = 10110 = 0 1 Proposition � n � For 0 ≤ k ≤ n we have = # P n , k . k

(d) Partitions.

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ).

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). Ex. Suppose λ = (3 , 2 , 2).

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2).

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) =

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns.

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = ⊆ 3 × 4: We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns.

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = ⊆ 3 × 4: We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns.

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = ⊆ 3 × 4: We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns. Let L n , k = { λ : λ ⊆ k × ( n − k ) } .

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = ⊆ 3 × 4: We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns. Let L n , k = { λ : λ ⊆ k × ( n − k ) } . There is a bijection g : L n , k → P n , k :

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then λ = (3 , 2 , 2) = ⊆ 3 × 4: We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns. Let L n , k = { λ : λ ⊆ k × ( n − k ) } . There is a bijection g : L n , k → P n , k : given λ ⊆ k × ( n − k ), P = g ( λ ) is formed by going from the SW corner of the rectangle to the NE corner along the rectangle and the SE boundary of λ .

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then g λ = (3 , 2 , 2) = ⊆ 3 × 4: ← → We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns. Let L n , k = { λ : λ ⊆ k × ( n − k ) } . There is a bijection g : L n , k → P n , k : given λ ⊆ k × ( n − k ), P = g ( λ ) is formed by going from the SW corner of the rectangle to the NE corner along the rectangle and the SE boundary of λ .

(d) Partitions. An (integer) partition is a weakly decreasing sequence of positive integers λ = ( λ 1 , . . . , λ m ). The associated Ferrers diagram has m left-justified rows of boxes with λ i boxes in row i Ex. Suppose λ = (3 , 2 , 2). Then g λ = (3 , 2 , 2) = ⊆ 3 × 4: ← → We say λ fits in a k × l rectangle , λ ⊆ k × l , if its Ferrers diagram has at most k rows and at most l columns. Let L n , k = { λ : λ ⊆ k × ( n − k ) } . There is a bijection g : L n , k → P n , k : given λ ⊆ k × ( n − k ), P = g ( λ ) is formed by going from the SW corner of the rectangle to the NE corner along the rectangle and the SE boundary of λ . Proposition � n � For 0 ≤ k ≤ n we have = # L n , k . k

Outline The theme Variation 1: binomial coefficients Variation 2: q -binomial coefficients Variation 3: fibonomial coefficients Coda: an open question and bibliography

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O .

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 .

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 .

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 . Note that n � �� � [ n ] | q =1 = 1 + 1 + · · · + 1 = n .

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 . Note that n � �� � [ n ] | q =1 = 1 + 1 + · · · + 1 = n . A q-factorial is [ n ]! = [1][2] · · · [ n ].

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 . Note that n � �� � [ n ] | q =1 = 1 + 1 + · · · + 1 = n . A q-factorial is [ n ]! = [1][2] · · · [ n ]. For 0 ≤ k ≤ n , the q-binomial coefficients or Gaussian polynomials are � n � [ n ]! = [ k ]![ n − k ]! . k

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 . Note that n � �� � [ n ] | q =1 = 1 + 1 + · · · + 1 = n . A q-factorial is [ n ]! = [1][2] · · · [ n ]. For 0 ≤ k ≤ n , the q-binomial coefficients or Gaussian polynomials are � n � [ n ]! = [ k ]![ n − k ]! . k Ex. We have � 4 � [2]![2]! = [4][3] [4]! [2][1] = 1 + q + 2 q 2 + q 3 + q 4 . = 2

A q-analogue of a mathematical object O (number, definition, theorem) is an object O ( q ) with O (1) = O . The standard q-analogue of n ∈ N is the polynomial [ n ] = 1 + q + q 2 + · · · + q n − 1 . Ex. We have [4] = 1 + q + q 2 + q 3 . Note that n � �� � [ n ] | q =1 = 1 + 1 + · · · + 1 = n . A q-factorial is [ n ]! = [1][2] · · · [ n ]. For 0 ≤ k ≤ n , the q-binomial coefficients or Gaussian polynomials are � n � [ n ]! = [ k ]![ n − k ]! . k Ex. We have � 4 � [2]![2]! = [4][3] [4]! [2][1] = 1 + q + 2 q 2 + q 3 + q 4 . = 2 � n � Note that it is not clear from the definition that is always in k N [ q ], the set of polynomials in q with coefficients in N .

Here is a q -analogue for the boundary conditions and recurrence relation for the binomial coefficients.

Here is a q -analogue for the boundary conditions and recurrence relation for the binomial coefficients. Theorem � n � n � � The q-binomial coefficients satisfy = = 1 and, for 0 n 0 < k < n, � n − 1 � � n − 1 � � n � q k = + k k k − 1

Here is a q -analogue for the boundary conditions and recurrence relation for the binomial coefficients. Theorem � n � n � � The q-binomial coefficients satisfy = = 1 and, for 0 n 0 < k < n, � n − 1 � � n − 1 � � n � q k = + k k k − 1 � n − 1 � � n − 1 � + q n − k = . k k − 1

Here is a q -analogue for the boundary conditions and recurrence relation for the binomial coefficients. Theorem � n � n � � The q-binomial coefficients satisfy = = 1 and, for 0 n 0 < k < n, � n − 1 � � n − 1 � � n � q k = + k k k − 1 � n − 1 � � n − 1 � + q n − k = . k k − 1 Since sums and products of elements of N [ q ] are again in N [ q ], we immediately get the following result. Corollary � n � For all 0 ≤ k ≤ n we have ∈ N [ q ] . k

(a) Words.

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } .

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) }

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) }

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4.

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4. Consider the inversion generating function � q inv w . I n , k ( q ) = w ∈W n , k

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4. Consider the inversion generating function � q inv w . I n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4. Consider the inversion generating function � q inv w . I n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 1 q 2 q 2 q 3 q 4 I 4 , 2 ( q ) = + + + + +

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4. Consider the inversion generating function � q inv w . I n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 1 q 2 q 2 q 3 q 4 I 4 , 2 ( q ) = + + + + + � � 4 = . 2

(a) Words. If w = a 1 . . . a n is a word over N then the inversion set of w is Inv w = { ( i , j ) : i < j and a i > a j } . The corresponding inversion number is inv w = # Inv w . Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then Inv w = { (1 , 2) , (1 , 5) , (3 , 5) , (4 , 5) } and inv w = 4. Consider the inversion generating function � q inv w . I n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 1 q 2 q 2 q 3 q 4 I 4 , 2 ( q ) = + + + + + � � 4 = . 2 Theorem � n � For all 0 ≤ k ≤ n we have = I n , k ( q ) . k

(b) Even more words.

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5.

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5. Consider the major index generating function � q maj w . M n , k ( q ) = w ∈W n , k

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5. Consider the major index generating function � q maj w . M n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5. Consider the major index generating function � q maj w . M n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 2 q 3 q 4 q 2 M 4 , 2 ( q ) = + + + q + +

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5. Consider the major index generating function � q maj w . M n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 2 q 3 q 4 q 2 M 4 , 2 ( q ) = + + + q + + � � 4 = . 2

(b) Even more words. If w = a 1 . . . a n is a word over N then the major index of w is � maj w = i . a i > a i +1 Ex. If w = a 1 a 2 a 3 a 4 a 5 = 10110 then a 1 > a 2 and a 4 > a 5 so maj w = 1 + 4 = 5. Consider the major index generating function � q maj w . M n , k ( q ) = w ∈W n , k Ex. When n = 4 and k = 2, W 4 , 2 : 0011 0101 0110 1001 1010 1100 q 0 q 2 q 3 q 4 q 2 M 4 , 2 ( q ) = + + + q + + � � 4 = . 2 Theorem � n � For all 0 ≤ k ≤ n we have = M n , k ( q ) . k

(c) Partitions.

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m .

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12.

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram.

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w .

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w .

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 01 1 1

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 01 1 1

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 1 1 0 1

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 1 1 0 1

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 01 1 1

(c) Partitions. If λ = ( λ 1 , . . . , λ m ) is a partition then its size is | λ | = λ 1 + · · · + λ m . Ex. If λ = (4 , 3 , 3 , 2) then | λ | = 4 + 3 + 3 + 2 = 12. Note that | λ | is the number of squares in its Ferrers diagram. Consider the size generating function � q | λ | . S n , k ( q ) = λ ∈L n , k Composing g : L n , k → P n , k and f : P n , k → W n , k gives a bijection h = f ◦ g : L n , k → W n , k such that, if h ( λ ) = w then | λ | = inv w . In fact, squares of λ correspond bijectively to elements of Inv w . Ex. When n = 5 and k = 2 0 h w = 10110 ← → λ = 01 1 1 Theorem � n � For all 0 ≤ k ≤ n we have = S n , k ( q ) . k

(d) Subspaces.

(d) Subspaces. Let q be a prime power and F q be the Galois field with q elements.

(d) Subspaces. Let q be a prime power and F q be the Galois field with q elements. Consider the n -dimensional vector space F n q .

(d) Subspaces. Let q be a prime power and F q be the Galois field with q elements. Consider the n -dimensional vector space F n q . Let V n , k ( q ) = { W : W is a k -dimensional subspace of F n q } .

(d) Subspaces. Let q be a prime power and F q be the Galois field with q elements. Consider the n -dimensional vector space F n q . Let V n , k ( q ) = { W : W is a k -dimensional subspace of F n q } . Ex. Let q = 3.