Counting of functions under (twelve)different restrictions f:N→M (NI=n,MI =m) 12=(2×2)×3 4口·￥①，43，t夏，里Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting farch9.202212/35
Counting # of functions under (twelve) different restrictions f : N → M (|N| = n, |M| = m) 12 = (2 × 2) × 3 Elements of N Elements of M Any f Injective f Surjective f distinguishable distinguishable indistinguishable distinguishable distinguishable indistinguishable indistinguishable indistinguishable Table: The Twelvefold Way (Functions). distinguishable vs. indistinguishable Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 12 / 35
Counting of functions under (twelve)different restrictions f:N→M (NI=n,MI=m) 12=(2×2)×3 Elements of N Elements of M Any f Injective f Surjective distinguishable distinguishable indistinguishable distinquishable distinguishable indistinguishable indistinguishable indistinguishable Table:The Twelvefold Way (Functions). 4口·￥①，43，t夏，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 12/35
Counting # of functions under (twelve) different restrictions f : N → M (|N| = n, |M| = m) 12 = (2 × 2) × 3 Elements of N Elements of M Any f Injective f Surjective f distinguishable distinguishable indistinguishable distinguishable distinguishable indistinguishable indistinguishable indistinguishable Table: The Twelvefold Way (Functions). distinguishable vs. indistinguishable Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 12 / 35
Counting of functions under (twelve)different restrictions f:N→M (NI=n,MI=m) 12=(2×2)×3 Elements of N Elements of M Any f Injective f Surjective distinguishable distinguishable indistinguishable distinquishable distinguishable indistinguishable indistinguishable indistinguishable Table:The Twelvefold Way (Functions). distinguishable vs.indistinguishable Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 12/35
Counting # of functions under (twelve) different restrictions f : N → M (|N| = n, |M| = m) 12 = (2 × 2) × 3 Elements of N Elements of M Any f Injective f Surjective f distinguishable distinguishable indistinguishable distinguishable distinguishable indistinguishable indistinguishable indistinguishable Table: The Twelvefold Way (Functions). distinguishable vs. indistinguishable Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 12 / 35
Balls Bins unrestricted ≤1≥1 unlabeled unlabeled labeled unlabeled unlabeled labeled labeled labeled Table:The Twelvefold Way (Balls into Bins Model). 4口·￥①，43，t夏，里Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 13/35
Balls Bins unrestricted ≤ 1 ≥ 1 unlabeled unlabeled labeled unlabeled unlabeled labeled labeled labeled Table: The Twelvefold Way (Balls into Bins Model). labeled vs. unlabeled Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 13 / 35
Balls Bins unrestricted ≤1≥1 unlabeled unlabeled labeled unlabeled unlabeled labeled labeled labeled Table:The Twelvefold Way (Balls into Bins Model). labeled us.unlabeled 4口·￥①，43，t夏，里Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 13/35
Balls Bins unrestricted ≤ 1 ≥ 1 unlabeled unlabeled labeled unlabeled unlabeled labeled labeled labeled Table: The Twelvefold Way (Balls into Bins Model). labeled vs. unlabeled Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 13 / 35