Gaussian binomial coefficients with negative arguments International Conference on Mathematical Analysis and Applications National Institute of Technology Jamshedpur Armin Straub November 2, 2020 University of South Alabama includes joint work with: Sam Formichella (University of South Alabama) Gaussian binomial coefficients with negative arguments Armin Straub 1 / 34
Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: Gaussian binomial coefficients with negative arguments Armin Straub 2 / 34
Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! ( q ; q ) n • q -binomial: D1 = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q Gaussian binomial coefficients with negative arguments Armin Straub 2 / 34
Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! ( q ; q ) n • q -binomial: D1 = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q � 6 � EG = 6 · 5 = 3 · 5 2 2 = (1 + q + q 2 + q 3 + q 4 + q 5 )(1 + q + q 2 + q 3 + q 4 ) � 6 � 2 1 + q q (1 + q + q 2 + q 3 + q 4 ) = (1 − q + q 2 ) (1 + q + q 2 ) � �� � � �� � =[3] q =[5] q Gaussian binomial coefficients with negative arguments Armin Straub 2 / 34
Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! ( q ; q ) n • q -binomial: D1 = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q � 6 � EG = 6 · 5 = 3 · 5 2 2 = (1 + q + q 2 + q 3 + q 4 + q 5 )(1 + q + q 2 + q 3 + q 4 ) � 6 � 2 1 + q q (1 + q + q 2 + q 3 + q 4 ) = (1 − q + q 2 ) (1 + q + q 2 ) � �� � � �� � � �� � Φ 6 (1) = 1 =Φ 6 ( q ) =[3] q =[5] q becomes invisible Gaussian binomial coefficients with negative arguments Armin Straub 2 / 34
Cyclotomic polynomials q -binomial coefficients The n th cyclotomic polynomial: DEF � ( q − ζ k ) where ζ = e 2 πi/n Φ n ( q ) = 1 � k<n ( k,n )=1 irreducible polynomial (nontrivial; Gauss!) with integer coefficients • [ n ] q = q n − 1 � q − 1 = Φ d ( q ) For primes: [ p ] q = Φ p ( q ) 1 <d � n d | n EG Φ 5 ( q ) = q 4 + q 3 + q 2 + q + 1 Φ 21 ( q ) = q 12 − q 11 + q 9 − q 8 + q 6 − q 4 + q 3 − q + 1 Φ 105 ( q ) = q 48 + q 47 + q 46 − q 43 − q 42 − 2 q 41 − q 40 − q 39 + q 36 + q 35 + q 34 + q 33 + q 32 + q 31 − q 28 − q 26 − q 24 − q 22 − q 20 + q 17 + q 16 + q 15 + q 14 + q 13 + q 12 − q 9 − q 8 − 2 q 7 − q 6 − q 5 + q 2 + q + 1 Gaussian binomial coefficients with negative arguments Armin Straub 3 / 34
q -binomials: factored and expanded q -binomial coefficients LEM n � n � [ n ] q ! Φ d ( q ) ⌊ n/d ⌋ − ⌊ k/d ⌋ − ⌊ ( n − k ) /d ⌋ � factored = [ k ] q ![ n − k ] q ! = k ∈ { 0 , 1 } q d =2 proof n n � � � Φ d ( q ) ⌊ n/d ⌋ [ n ] q ! = Φ d ( q ) = m =1 d =2 d | m d> 1 • In particular, the q -binomial is a polynomial. (of degree k ( n − k ) ) Gaussian binomial coefficients with negative arguments Armin Straub 4 / 34
q -binomials: factored and expanded q -binomial coefficients LEM n � n � [ n ] q ! Φ d ( q ) ⌊ n/d ⌋ − ⌊ k/d ⌋ − ⌊ ( n − k ) /d ⌋ � factored = [ k ] q ![ n − k ] q ! = k ∈ { 0 , 1 } q d =2 proof n n � � � Φ d ( q ) ⌊ n/d ⌋ [ n ] q ! = Φ d ( q ) = m =1 d =2 d | m d> 1 • In particular, the q -binomial is a polynomial. (of degree k ( n − k ) ) � 6 � EG = q 8 + q 7 + 2 q 6 + 2 q 5 + 3 q 4 + 2 q 3 + 2 q 2 + q + 1 expanded 2 q � 9 � = q 18 + q 17 + 2 q 16 + 3 q 15 + 4 q 14 + 5 q 13 + 7 q 12 3 q + 7 q 11 + 8 q 10 + 8 q 9 + 8 q 8 + 7 q 7 + 7 q 6 + 5 q 5 + 4 q 4 + 3 q 3 + 2 q 2 + q + 1 • The coefficients are positive and unimodal . Sylvester, 1878 Gaussian binomial coefficients with negative arguments Armin Straub 4 / 34
q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q Gaussian binomial coefficients with negative arguments Armin Straub 5 / 34
q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q � n � The coefficient of q m in q counts the number of k • k -element subsets of n whose normalized sum is m , Gaussian binomial coefficients with negative arguments Armin Straub 5 / 34
q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q � n � The coefficient of q m in q counts the number of k • k -element subsets of n whose normalized sum is m , • partitions λ of m whose Ferrer’s diagram fits in a k × ( n − k ) box. Gaussian binomial coefficients with negative arguments Armin Straub 5 / 34
q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q Gaussian binomial coefficients with negative arguments Armin Straub 6 / 34
q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q � n � THM = number of k -dim. subspaces of F n D4 q k q Gaussian binomial coefficients with negative arguments Armin Straub 6 / 34
q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q � n � THM = number of k -dim. subspaces of F n D4 q k q THM Suppose yx = qxy (and that q commutes with x, y ). Then: n � n � � ( x + y ) n = x k y n − k D5 k q k =0 Gaussian binomial coefficients with negative arguments Armin Straub 6 / 34
q -calculus q -binomial coefficients The q -derivative : DEF D q f ( x ) = f ( qx ) − f ( x ) qx − x D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x Gaussian binomial coefficients with negative arguments Armin Straub 7 / 34
q -calculus q -binomial coefficients • D q e x q = e x The q -derivative : DEF q • e x q · e y q = e x + y D q f ( x ) = f ( qx ) − f ( x ) q provided that yx = qxy qx − x q · e − x • e x 1 /q = 1 D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x ∞ x n ∞ ( x (1 − q )) n 1 � � • The q -exponential : e x q = [ n ] q ! = = ( q ; q ) n ( x (1 − q ); q ) ∞ n =0 n =0 Gaussian binomial coefficients with negative arguments Armin Straub 7 / 34
q -calculus q -binomial coefficients • D q e x q = e x The q -derivative : DEF q • e x q · e y q = e x + y D q f ( x ) = f ( qx ) − f ( x ) q provided that yx = qxy qx − x q · e − x • e x 1 /q = 1 D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x ∞ x n ∞ ( x (1 − q )) n 1 � � • The q -exponential : e x q = [ n ] q ! = = ( q ; q ) n ( x (1 − q ); q ) ∞ n =0 n =0 • The q -integral : from formally inverting D q � x ∞ � q n xf ( q n x ) f ( x ) d q x := (1 − q ) 0 n =0 Gaussian binomial coefficients with negative arguments Armin Straub 7 / 34
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