Counting Counting of functions under (twelve)different restrictions 4口·￥①，4三，t更，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 11/35
tuples permutations combinations Counting compositions set partitions integer partitions Counting # of functions under (twelve) different restrictions Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 11 / 35
Counting tuples permutations combinations Counting of functions under (twelve)different restrictions 4口·￥①，4三，t更，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 11/35
tuples permutations combinations Counting compositions set partitions integer partitions Counting # of functions under (twelve) different restrictions Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 11 / 35
Counting tuples compositions permutations set partitions combinations integer partitions Counting of functions under (twelve)different restrictions 4口·￥①，43，t夏，里)Q0 Jun Ma (majunainju.edu.cn) 2-3 Counting March 9.2022 11/35
tuples permutations combinations Counting compositions set partitions integer partitions Counting # of functions under (twelve) different restrictions Jun Ma (majun@nju.edu.cn) 2-3 Counting March 9, 2022 11 / 35
Counting of functions under (twelve)different restrictions 4口·￥①，4三，t更，3)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 NM (INI n,MI =m) 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