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Next: 3 ¿¹à¼°¤Îʸ»ú¤ÎÃÖ´¹ Up: 4 ¹ÔÎó¼° Previous: 1 ¹ÔÎó¼°¤ÎƳ½Ð   Contents

2 ÃÖ´¹

ÄêµÁ 4.1 (ʸ»ú¤ÎÃÖ´¹)   $ n$ ¸Ä¤Îʸ»ú $ \{1,2,\cdots,n\}$ ¤«¤é ¼«Ê¬¼«¿È $ \{1,2,\cdots,n\}$ ¤Ø¤Î $ 1$ ÂÐ $ 1$ ¤Î¼ÌÁü¤ò $ n$ ʸ»ú¤ÎÃÖ´¹¡Êpermutation¡Ë¤È¤¤¤¦¡¥ $ n$ ʸ»ú¤ÎÃÖ´¹ $ \sigma$ ¤¬¼ÌÁü

$\displaystyle 1\to k_{1}\,,\quad 2\to k_{2}\,,\quad \cdots n\to k_{n}\,$ (553)

¤Î¤È¤­ $ \sigma$ ¤ò

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ k_{1} & k_{2} & \cdots & k_{n} \end{pmatrix}$ (554)

¤Èɽ¤ï¤¹¡¥ ¼ÌÁü $ j\to k_{j}$ ¤ò $ \sigma(j)=k_{j}$ ¤Èɽ¤ï¤¹¡¥

Îã 4.2 (ÃÖ´¹¤Î¶ñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{pmatrix}\,,$ (555)
  $\displaystyle 1\to3\,,\quad 2\to1\,,\quad 4\to4\,,\quad 4\to2\,,$ (556)
  $\displaystyle \sigma(1)=3\,,\quad \sigma(2)=1\,,\quad \sigma(4)=4\,,\quad \sigma(4)=2\,.$ (557)

Îã 4.3 (ÃÖ´¹¤Îɽµ­)  

$\displaystyle \begin{pmatrix}1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 1 \end{pmatrix}= \beg...
...4 & 1 & 3 \end{pmatrix}= \begin{pmatrix}3 & 1 & 4 \\ 4 & 3 & 1 \end{pmatrix}\,.$ (558)

Ʊ¤¸¿ô»ú¤ËÃÖ´¹¤¹¤ë¾ì¹ç¤Ï¾Êά²Äǽ¡¥ ʤ٤ë½ç¤Ï¤É¤¦¤Ç¤âÎɤ¤¡¥

ÄêµÁ 4.4 (ÃÖ´¹¤ÎÀÑ)   Æó¤Ä¤ÎÃÖ´¹

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ k_{1} & k_{2} & \cdots & k_{n} \end{pmatrix}\,,$ (559)
$\displaystyle \tau$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ l_{1} & l_{2} & \cdots & l_{n} \end{pmatrix}$ (560)

¤ÎÀÑ $ \sigma\tau$ ¤ò

$\displaystyle \sigma\tau$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ k_{1} & k_{2} & \cdots & k...
...rix}1 & 2 & \cdots & n \\ k_{l_1} & k_{l_2} & \cdots & k_{l_n} \end{pmatrix}\,,$ (561)

¤Þ¤¿¤Ï

$\displaystyle (\sigma\tau)(i)=\sigma(\tau(i))\,,\quad i=1,2,\cdots,n$ (562)

¤ÈÄêµÁ¤¹¤ë¡¥

Îã 4.5 (ÃÖ´¹¤ÎÀѤζñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{pmatrix}\,,\quad \tau= \begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{pmatrix}\,,$ (563)
$\displaystyle \sigma\tau$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{pmatrix} \begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{pmatrix}$ (564)
  $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ \sigma(\tau(1)) & \sigma(\tau(2...
...ix}1 & 2 & 3 & 4 \\ \sigma(2) & \sigma(3) & \sigma(4) & \sigma(1) \end{pmatrix}$ (565)
  $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 3 & 1 & 2 & 4 \end{pmatrix}\,,$ (566)
$\displaystyle \tau\sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{pmatrix} \begin{pmatrix}1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{pmatrix}$ (567)
  $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ \tau(\sigma(1)) & \tau(\sigma(2...
...in{pmatrix}1 & 2 & 3 & 4 \\ \tau(4) & \tau(3) & \tau(1) & \tau(2) \end{pmatrix}$ (568)
  $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 1 & 4 & 2 & 3 \end{pmatrix}\,.$ (569)

Ãí°Õ 4.6 (ÃÖ´¹¤ÎÀѤÏÈó²Ä´¹)   °ìÈÌŪ¤Ë $ \sigma\tau=\tau\sigma$ ¤ÏÀ®Î©¤·¤Ê¤¤¡¥

ÄêµÁ 4.7 (ñ°ÌÃÖ´¹)   Á´¤Æ¤Îʸ»ú¤òÆ°¤«¤µ¤Ê¤¤ÃÖ´¹

$\displaystyle \epsilon$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ 1 & 2 & \cdots & n \end{pmatrix}$ (570)

¤òñ°ÌÃÖ´¹¤È¸Æ¤Ö¡¥

ÄêµÁ 4.8 (µÕÃÖ´¹)   ÃÖ´¹ $ \sigma$ ¤ËÂФ·¤Æ

$\displaystyle \sigma\tau=\tau\sigma=\epsilon$ (571)

¤òËþ¤¿¤¹ÃÖ´¹ $ \tau$ ¤ò $ \sigma$ ¤ÎµÕÃÖ´¹¤È¸Æ¤Ó¡¤ $ \tau=\sigma^{-1}$ ¤Èɽ¤ï¤¹¡¥

ÄêÍý 4.9 (µÕÃÖ´¹)   ÃÖ´¹

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & \cdots & n \\ k_{1} & k_{2} & \cdots & k_{n} \end{pmatrix}$ (572)

¤ÎµÕÃÖ´¹¤Ï

$\displaystyle \sigma^{-1}$ $\displaystyle = \begin{pmatrix}k_{1} & k_{2} & \cdots & k_{n} \\ 1 & 2 & \cdots & n \end{pmatrix}$ (573)

¤ÇÍ¿¤¨¤é¤ì¤ë¡¥

Îã 4.10 (µÕÃÖ´¹¤Î¶ñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 1 & 3 & 2 \end{pmatrix}\,,$ (574)
$\displaystyle \sigma^{-1}$ $\displaystyle = \begin{pmatrix}4 & 5 & 1 & 3 & 2 \\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}= \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 4 & 1 & 2 \end{pmatrix}\,.$ (575)

ÄêµÁ 4.11 (½ä²óÃÖ´¹)   $ n$ ¸Ä¤Îʸ»ú $ \{1,2,\cdots,n\}$ ¤Î¤¦¤Á $ r$ ¸Ä¤Îʸ»ú $ \{k_{1},k_{2},\cdots,k_{r}\}$ ¤Î¤ß¤ò $ k_{1}\to k_{2},k_{2}\to k_{3},\cdots,k_{r}\to k_{1}$ ¤È½ç¤Ë¤º¤é¤·¡¤ »Ä¤ê¤Îʸ»ú $ \{k_{r+1},\cdots,k_{n}\}$ ¤ò $ k_{r+1}\to k_{r+1},\cdots,k_{n}\to k_{n}$ ¤ÈÆ°¤«¤µ¤Ê¤¤ ¼ÌÁü¤ÎÃÖ´¹¤ò½ä²óÃÖ´¹¤È¤¤¤¦¡¥ ½ä²óÃÖ´¹¤Ï

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}k_{1} & k_{2} & \cdots & k_{r-1} & k_{r} & k_{r+...
...k_{2} & k_{3} & \cdots & k_{r} & k_{1} & k_{r+1} & \cdots & k_{n} \end{pmatrix}$ (576)
  $\displaystyle = \begin{pmatrix}k_{1} & k_{2} & \cdots & k_{r-1} & k_{r} \\ k_{2} & k_{3} & \cdots & k_{r} & k_{1} \end{pmatrix}$ (577)

¤Èɽ¤ï¤µ¤ì¡¤¾Êά¤¹¤ë¤È¤­¤Ï

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}k_{1} & k_{2} & \cdots & k_{r} \end{pmatrix}$ (578)

¤È½ñ¤¯¡¥

Îã 4.12 (½ä²óÃÖ´¹¤Î¶ñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 1 & 5 & 2 & 4 & 3 \end{pmat...
...\\ 5 & 2 & 3 \end{pmatrix}= \begin{pmatrix}2 & 5 & 3 \\ 5 & 3 & 2 \end{pmatrix}$ (579)
  $\displaystyle = \begin{pmatrix}2 & 5 & 3 \end{pmatrix}= \begin{pmatrix}5 & 3 & 2 \end{pmatrix}= \begin{pmatrix}3 & 2 & 5 \end{pmatrix}\,,$ (580)
$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 2 & 4 & 5 \end{pmat...
...\\ 3 & 1 & 2 \end{pmatrix}= \begin{pmatrix}1 & 3 & 2 \\ 3 & 2 & 1 \end{pmatrix}$ (581)
  $\displaystyle = \begin{pmatrix}1 & 3 & 2 \end{pmatrix}= \begin{pmatrix}3 & 2 & 1 \end{pmatrix}= \begin{pmatrix}2 & 1 & 3 \end{pmatrix}\,.$ (582)

ÄêÍý 4.13 (ÃÖ´¹¤ò½ä²óÃÖ´¹¤ÎÀѤÇɽ¤ï¤¹)   Ǥ°Õ¤ÎÃÖ´¹ $ \sigma$ ¤Ï½ä²óÃÖ´¹ $ \sigma_{1},\sigma_{2},\cdots,\sigma_{n}$ ¤ÎÀÑ¤Ç $ \sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{n}$ ¤Èɽ¤ï¤µ¤ì¤ë¡¥

Îã 4.14 (ÃÖ´¹¤ò½ä²óÃÖ´¹¤ÎÀѤÇɽ¤ï¤¹·×»»Îã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 1 & 6 & 2 & 7 &...
...in{pmatrix}1 & 4 & 2 & 3 & 6 & 5 & 7 \\ 4 & 2 & 1 & 6 & 5 & 7 & 3 \end{pmatrix}$ (583)
  $\displaystyle = \begin{pmatrix}1 & 4 & 2 & 3 & 6 & 5 & 7 \\ 4 & 2 & 1 & 3 & 6 &...
...2 & 1 \end{pmatrix} \begin{pmatrix}3 & 6 & 5 & 7 \\ 6 & 5 & 7 & 3 \end{pmatrix}$ (584)
  $\displaystyle = \begin{pmatrix}1 & 4 & 2 \end{pmatrix} \begin{pmatrix}3 & 6 & 5 & 7 \end{pmatrix}\,.$ (585)

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 3 & 7 & 1 & 4 &...
...in{pmatrix}1 & 5 & 4 & 2 & 3 & 7 & 6 \\ 5 & 4 & 1 & 3 & 7 & 6 & 2 \end{pmatrix}$ (586)
  $\displaystyle = \begin{pmatrix}1 & 5 & 4 & 2 & 3 & 7 & 6 \\ 5 & 4 & 1 & 2 & 3 &...
...4 & 1 \end{pmatrix} \begin{pmatrix}2 & 3 & 7 & 6 \\ 3 & 7 & 6 & 2 \end{pmatrix}$ (587)
  $\displaystyle = \begin{pmatrix}1 & 5 & 4 \end{pmatrix} \begin{pmatrix}2 & 3 & 7 & 6 \end{pmatrix}\,.$ (588)

ÄêµÁ 4.15 (¸ß´¹)   $ 2$ ʸ»ú¤Î½ä²óÃÖ´¹ $ (i\quad j)$ ¤ò ¸ß´¹¤È¤¤¤¦¡¥

ÄêÍý 4.16 (½ä²óÃÖ´¹¤ò¸ß´¹¤ÎÀѤÇɽ¤ï¤¹)   Ǥ°Õ¤Î½ä²óÃÖ´¹¤Ï¸ß´¹¤ÎÀѤÇɽ¤ï¤µ¤ì¤ë¡¥ ¤¿¤È¤¨¤Ð¡¤¤½¤Î°ì¤Ä¤È¤·¤Æ

$\displaystyle \begin{pmatrix}k_1 & k_2 & \cdots & k_r \end{pmatrix}= \begin{pma...
...s \begin{pmatrix}k_1 & k_3 \end{pmatrix} \begin{pmatrix}k_1 & k_2 \end{pmatrix}$ (589)

¤Èɽ¤ï¤µ¤ì¤ë¡¥

Îã 4.17 (ÃÖ´¹¤ò¸ß´¹¤ÎÀѤÇɽ¤ï¤¹)   ¤¿¤È¤¨¤Ð

$\displaystyle \begin{pmatrix}1 & 2 & 3 & 4 \end{pmatrix}$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \\ 4 & 2 & 3 & 1 \end{pmatrix} \be...
...1 & 4 \end{pmatrix} \begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{pmatrix}$ (590)
  $\displaystyle = \begin{pmatrix}1 & 4 \end{pmatrix} \begin{pmatrix}1 & 3 \end{pmatrix} \begin{pmatrix}1 & 2 \end{pmatrix}\,.$ (591)

¤¿¤È¤¨¤Ð

$\displaystyle \begin{pmatrix}1 & 2 & 3 & 4 \end{pmatrix}$ $\displaystyle = \begin{pmatrix}1 & 3 \end{pmatrix} \begin{pmatrix}1 & 4 \end{pm...
...atrix} \begin{pmatrix}2 & 3 \end{pmatrix} \begin{pmatrix}1 & 3 \end{pmatrix}\,.$ (592)

Ãí°Õ 4.18   ¸ß´¹¤ÎÀѤÇɽ¤ï¤¹ÊýË¡¤Ï´öÄ̤ê¤â¤¢¤ë¡¥

ÄêµÁ 4.19 (ÃÖ´¹¤ÎÉä¹æ)   ÃÖ´¹ $ \sigma$ ¤¬ $ m$ ¸Ä¤Î¸ß´¹¤ÎÀѤÇɽ¤ï¤µ¤ì¤ë¤È¤­ $ \sigma$ ¤ÎÉä¹æ¡Êsign¡Ë¤ò

$\displaystyle \mathrm{sgn}\,(\sigma)=(-1)^{m}$ (593)

¤ÈÄêµÁ¤¹¤ë¡¥

Îã 4.20 (ÃÖ´¹¤ÎÉä¹æ¤Î¶ñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \end{pmatrix}= \begin{pmatrix}1 & ...
...{pmatrix} \begin{pmatrix}1 & 3 \end{pmatrix} \begin{pmatrix}1 & 2 \end{pmatrix}$ (594)

¤è¤ê

$\displaystyle \mathrm{sgn}\,(\sigma)$ $\displaystyle = (-1)^{3}=-1$ (595)

¤È¤Ê¤ë¡¥¤Þ¤¿

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 \end{pmatrix}= \begin{pmatrix}1 & ...
...{pmatrix} \begin{pmatrix}2 & 3 \end{pmatrix} \begin{pmatrix}1 & 3 \end{pmatrix}$ (596)

¤è¤ê

$\displaystyle \mathrm{sgn}\,(\sigma)$ $\displaystyle = (-1)^{5}=-1$ (597)

¤Ç¤¢¤ë¡¥

ÄêÍý 4.21 (ÃÖ´¹¤ÎÉä¹æ¤Î°ì°ÕÀ­)   ÃÖ´¹ $ \sigma$ ¤ÎÉä¹æ $ \mathrm{sgn}\,(\sigma)$ ¤Ï ¸ß´¹¤ÎÀѤÎɽ¤ï¤·Êý¤Ë¤è¤é¤º°ì°Õ¤ËÄê¤Þ¤ë¡¥

ÄêÍý 4.22 (ÃÖ´¹¤ÎÉä¹æ¤ÎÀ­¼Á)  

  $\displaystyle \mathrm{sgn}\,(\epsilon)=1$ (598)
  $\displaystyle \mathrm{sgn}\,(\sigma\tau)=\mathrm{sgn}\,(\sigma)\mathrm{sgn}\,(\tau)$ (599)
  $\displaystyle \mathrm{sgn}\,(\sigma^{-1})=\mathrm{sgn}\,(\sigma)$ (600)

ÄêµÁ 4.23 (¶öÃÖ´¹¡¤´ñÃÖ´¹)   $ \mathrm{sgn}\,(\sigma)=1$ ¤È¤Ê¤ëÃÖ´¹¤ò ¶öÃÖ´¹¤È¸Æ¤Ó¡¤ $ \mathrm{sgn}\,(\sigma)=-1$ ¤È¤Ê¤ëÃÖ´¹¤ò ´ñÃÖ´¹¤È¸Æ¤Ö¡¥

Îã 4.24 (¶öÃÖ´¹¡¤´ñÃÖ´¹¤Î¶ñÂÎÎã)  

$\displaystyle \sigma$ $\displaystyle = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 6 & 8 &...
... & 9 & 5 & 2 & 6 & 4 & 3 & 3 \\ 7 & 9 & 5 & 1 & 6 & 4 & 2 & 8 & 2 \end{pmatrix}$ (601)
  $\displaystyle = \begin{pmatrix}1 & 7 & 9 & 5 \\ 7 & 9 & 5 & 1 \end{pmatrix} \be...
...trix} \begin{pmatrix}2 & 6 & 4 \end{pmatrix} \begin{pmatrix}3 & 8 \end{pmatrix}$ (602)
  $\displaystyle = \begin{pmatrix}1 & 5 \end{pmatrix} \begin{pmatrix}1 & 9 \end{pm...
...{pmatrix} \begin{pmatrix}2 & 6 \end{pmatrix} \begin{pmatrix}3 & 8 \end{pmatrix}$ (603)

¤è¤ê

$\displaystyle \mathrm{sgn}\,(\sigma)$ $\displaystyle =(-1)^6=1$ (604)

¤È¤Ê¤ë¡¥ $ \sigma$ ¤Ï¶öÃÖ´¹¤Ç¤¢¤ë¡¥

ÄêµÁ 4.25 (ÃÖ´¹Á´ÂΤν¸¹ç)   $ n$ ʸ»ú¤ÎÃÖ´¹ $ \sigma$ ¤ÎÁ´ÂΤν¸¹ç¤ò $ S_{n}$ ¤È½ñ¤¯¡¥

Ãí°Õ 4.26 (ÃÖ´¹Á´ÂΤν¸¹ç¤ÎÍ×ÁǤθĿô)   $ n$ ʸ»ú¤ÎÃÖ´¹¤Ï¼ÌÁü

$\displaystyle \{1,2,\cdots,n\}\quad\to\quad \{k_{1},k_{2},\cdots,k_{n}\}$ (605)

¤Ç¤¢¤ë¤«¤é¡¤ ¤½¤Î¸Ä¿ô¤Ï $ n$ ¸Ä¤Îʸ»ú¤Î½çÎóÁȹç¤ï¤»¤ËÅù¤·¤¤¡¥ ¤è¤Ã¤Æ½¸¹ç $ S_{n}$ ¤Î¸Ä¿ô¤Ï $ n!$ ¤Ç¤¢¤ë¡¥

Îã 4.27 (ÃÖ´¹Á´ÂΤν¸¹ç¤Î¶ñÂÎÎã)  

$\displaystyle S_{1}$ $\displaystyle =\left\{ \begin{pmatrix}1 \\ 1 \end{pmatrix} \right\}= \{\underset{\text{¶ö}}{\epsilon}\}$ (606)
$\displaystyle S_{2}$ $\displaystyle = \left\{ \begin{pmatrix}1 & 2 \\ 1 & 2 \end{pmatrix}, \begin{pma...
...ö}}{\epsilon},\quad \underset{\text{´ñ}}{\begin{pmatrix}1 & 2 \end{pmatrix}} \}$ (607)
$\displaystyle S_{3}$ $\displaystyle = \left\{ \begin{pmatrix}1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}, \b...
...& 2 \end{pmatrix}, \begin{pmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix} \right\}$ (608)
  $\displaystyle = \{ \underset{\text{¶ö}}{\epsilon},\quad \underset{\text{´ñ}}{\b...
...end{pmatrix}},\quad \underset{\text{´ñ}}{\begin{pmatrix}1 & 2 \end{pmatrix}} \}$ (609)

Ìä 4.28 (ÃÖ´¹Á´ÂΤν¸¹ç)   $ 4$ ¼¡¤ÎÃÖ´¹Á´ÂΤν¸¹ç $ S_4$ ¤ÎÍ×ÁÇÁ´¤Æ¤ò½ñ¤­½Ð¤»¡¥ ¤Þ¤¿¤½¤Î¶ö´ñ¤â½Ò¤Ù¤è¡¥

Ìä 4.29 (¶öÃÖ´¹¡¤´ñÃÖ´¹¤Î¸Ä¿ô)   $ S_{n}$ $ (n\geq2)$ ¤Ë´Þ¤Þ¤ì¤ë¶öÃÖ´¹¤È´ñÃÖ´¹¤Î¸Ä¿ô¤ÏÅù¤·¤¤¡¥ ¤³¤ì¤ò¼¨¤»¡¥


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Created at 2004/11/26