1 . 在数字
的任意一个排列
:
中,如果对于
,
,有
,那么就称
为一个逆序对.记排列
中逆序对的个数为
.如
时,在排列
:3,2,4,1中,逆序对有
,
,
,
,则
.
(1)设排列
:
,写出两组具体的排列
,分别满足:①
,②
;
(2)对于数字1,2,…,n的一切排列
,求所有
的算术平均值;
(3)如果把排列A:
中两个数字
交换位置,而其余数字的位置保持不变,那么就得到一个新的排列,
:
,求证:
为奇数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed5929d3faf0bb567573cc46fabd30aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6d99dfbc36d9be71895dfdb732f321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52f2c7c9305b404f7363a376af101aa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad7ae1214cc78e72fb613d7e649bc27b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48a1d999ab14d86a73dd17df33f23b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bba7a40102e81f5759f7b05ebf5a18c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/990eaf5dbba84f199bdc438da81fcfa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0481d24e2af1e0cd348732b9444d1dde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad20e2bc6576fc461419f8f138d26e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce2830528c2ea6b5d4df0c77644e33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b08a341f6e37e92710bf95531217e6.png)
(1)设排列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec99c57bf7997bd93e1ed8f48d5af9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c42030a33382b003b81c2b148a216fe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb79a2396402d62e725578f015e5c06a.png)
(2)对于数字1,2,…,n的一切排列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bba7a40102e81f5759f7b05ebf5a18c.png)
(3)如果把排列A:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61e68f643eb061dea7def83952c1f56c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e3d172f08313520e76b6cbc2ff9980c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c687695f558d18f8b220be874a723170.png)
您最近一年使用:0次
名校
2 . 已知
,对于有限集
,令
表示集合
中元素的个数.例如:当
时,
,
.
(1)当
时,请直接写出集合
的子集的个数;
(2)当
时,
,
都是集合
的子集(
,
可以相同),并且
.求满足条件的有序集合对
的个数;
(3)假设存在集合
、
具有以下性质:将1,1,2,2,··,
,
.这
个整数按某种次序排成一列,使得在这个序列中,对于任意
,
与
之间恰好排列
个整数.证明:
是4的倍数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54d5262a2b9c5f9414e590a614cf4c6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe2ecc1431daa3d31c1fa94d0c38b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9387062f286152ca34cc40621fb5a5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cecd12504283118ed6bef7eafa123c4e.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9b6d21c9d991e18d644b72afba4e88b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41be4f4fbf21555f325caf280c392c00.png)
(3)假设存在集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe2ecc1431daa3d31c1fa94d0c38b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe2ecc1431daa3d31c1fa94d0c38b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f3d9e6b9e450e436227d6b2e2a3e08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f338dcaa1aebbfe470899cc2544edd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de05de2e0a7cccae0e2e39650385c34f.png)
您最近一年使用:0次
3 . 在数字1,2,…,n(n≥2)的任意一个排列A:a1,a2,,an中,如果对于i,j∈N*,i<j,有ai>aj,那么就称(ai,aj)为一个逆序对.记排列A中逆序对的个数为S(A).
如n=4时,在排列B:3,2,4,1中,逆序对有(3,2),(3,1),(2,1),(4,1),
则S(B)=4.
(1)设排列 C:3,5,6,4,1,2,写出S(C)的值;
(2)对于数字1,2,...,n的一切排列A,求所有S(A)的算术平均值;
(3)如果把排列A:a1,a2,...,an中两个数字ai,aj(i<j)交换位置,而其余数字的位置保持不变,那么就得到一个新的排列A':b1,b2,…,bn,求证:S(A)+S(A')为奇数.
如n=4时,在排列B:3,2,4,1中,逆序对有(3,2),(3,1),(2,1),(4,1),
则S(B)=4.
(1)设排列 C:3,5,6,4,1,2,写出S(C)的值;
(2)对于数字1,2,...,n的一切排列A,求所有S(A)的算术平均值;
(3)如果把排列A:a1,a2,...,an中两个数字ai,aj(i<j)交换位置,而其余数字的位置保持不变,那么就得到一个新的排列A':b1,b2,…,bn,求证:S(A)+S(A')为奇数.
您最近一年使用:0次
2016-12-04更新
|
637次组卷
|
2卷引用:北京市第二中学2022-2023学年高二上学期11月学段考试数学试题