Counting of functions under (twelve)different restrictions 4口·￥①，43，t夏，3)Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting March12,202013/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 13 / 34
Counting of functions under (twelve)different restrictions f NM (INI n,MI =m) 4口·￥①，43，t夏，里Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting March12,2020.13/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 13 / 34
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 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting March12,2020.13/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 13 / 34
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 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting March12,202013/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 13 / 34
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 3，t里，里)Q0 Hengfeng Wei (hfweixinju.edu.cn) 2-3 Counting March12,202013/34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Hengfeng Wei (hfwei@nju.edu.cn) 2-3 Counting March 12, 2020 13 / 34