T(n,k) 表示将 1～n 排成一列，恰有 k 个数字与其位置吻合的排列种数，问一些有关问题

luyuanhong

王守恩

Treenewbee 发表于 2023-12-9 01:18
你把原始递推公式代进去计算一下即可

一个很丑的公式：

\(y_{n,k}=\big(\frac{n![(n-k)!/E]}{k!(n-k)!}-\frac{(n+1)![(n-k)!/E]}{(k+1)!(n-k)!}-\frac{n![(n-k-1)!/E]}{(k+1)!(n-k-1)!}-[\frac{n!}{E*k!}]+[\frac{n!}{E(k+1)!}]+[\frac{(n+1)!}{E(k+1)!}]\big)(-1)^{n+k}\)

1. Table[((n! Round[(n - k)!/E])/( k! (n - k)!)
   
2. - ((n + 1)! Round[(n - k)!/E])/((k + 1)! (n - k)!)
   
3. - (n! Round[(n - k - 1)!/E])/((k + 1)! (n - k - 1)!)
   
4. -  Round[n!/(E k!)] + Round[n!/(E (k + 1)!)]
   
5. + Round[(n + 1)!/(E (k + 1)!)]) (-1)^(k + n), {n, 4, 19}, {k, 2,  n - 2}]

复制代码

{0},
{1, 1},
{1, 1, 0},
{1, 1, 1, 2},
{1, 2, 2, 2, 0},
{2, 2, 3, 3, 2, 1},
{1, 3, 5, 6, 5, 3, 1},
{2, 4, 7, 9, 8, 5, 3, 2},
{2, 4, 9, 13, 14, 13, 9, 4, 0},
{2, 6, 13, 20, 25, 24, 18, 10, 5, 2},
{2, 6, 15, 27, 40, 44, 37, 25, 13, 5, 1},
{3, 8, 20, 39, 60, 73, 72, 57, 34, 16, 6, 2},
{2, 9, 25, 53, 89, 120, 131, 115, 82, 46, 20, 7, 2},
{3, 10, 29, 69, 126, 188, 227, 223, 179, 116, 61, 25, 8, 2},
{3, 11, 37, 90, 177, 286, 376, 408, 364, 267, 161, 78, 30, 8, 2},
{3, 13, 43, 114, 243, 422, 602, 714, 705, 578, 391, 217, 98, 35, 10, 2}}

王守恩

T(n,k) 表示将 1～n 排成一列, 恰有 k 个数字与其位置吻合的排列种数。

\(T(n,k)=\frac{n!}{k! (n - k)!}*\big[\frac{(n - k)!}{e}\big],[\ \ ]\)表示四舍五入，e=2.7182818284590452....

\(\frac{n!}{k! (n - k)!}\)=杨辉三角, 是整数;  \(\big[\frac{(n - k)!}{e}\big]\)也是整数, 参考OEIS--A000166。

王守恩

还简单些(OEIS还没有这样组合的)。

T(n,k)=\(\frac{n![k!/e]}{k! (n - k)!}\)

{0, 1},
{0, 3, 2},
{0, 6, 8, 9},
{0, 10, 20, 45, 44},
{0, 15, 40, 135, 264, 265},
{0, 21, 70, 315, 924, 1855, 1854},
{0, 28, 112, 630, 2464, 7420, 14832, 14833},
{0, 36, 168, 1134, 5544, 22260, 66744, 133497, 133496},
{0, 45, 240, 1890, 11088, 55650, 222480, 667485, 1334960, 1334961},
{0, 55, 330, 2970, 20328, 122430, 611820, 2447445, 7342280, 14684571, 14684570},
{0, 66, 440, 4455, 34848, 244860, 1468368, 7342335, 29369120, 88107426, 176214840,176214841},

wintex 网友！你不感到有想学习的冲动吗？!  我用的就是网上免费下载的学习版。