DcN

DcN，是318`4创造的便于打字的记号，主要用于写FOS中作为序数的项，几乎不可能良定义，主要通过枚举列表来主观推算定义。DcN的极限是$\phi (10,0)$，Y_cpper将其扩展到${\mathrm{\Gamma }}_0$，现在一般在$\phi (\omega ,0)$之后使用这个扩展。DcN的所有符号都集中在电脑键盘左上方的一小片区域，qwerty13456789，456789使用频率很低，Veblen记号需要随时要输入括号下标希腊字母，1-Y则有比DcN更长的常见表达式，所以DcN便于打字，是专为FOS中常见序数设计的记号；但因为可读性差，DcN一直没能流传开来。

1-Y不必多说，是Yukito在2019年创造的序列记号，首次引入山脉图，在Y(1,3)之前有非常优秀的性质，因此也常用于表达FOS中作为项的序数。不过为简便，写1-Y表达式时通常省略Y( )外壳和逗号，超过10的数字用拉丁字母表示，超过36则无写法。

在研究FOS的定义、构造原理时，使用1-Y的情况更多，因为1-Y的山脉结构和FOS中项内部的山脉结构类似；而在研究FOS的分析、强度形成原理时，为简便起见，使用DcN更多。在阅读FOS的枚举列表之前，请务必将本文的枚举列表熟透于心中，否则用Veblen记号或OCF写FOS表达式费时费力。

序数（十进制+Cantor记号+Veblen记号+BOCF）	DcN	1-Y
$0$	0	$\varnothing$
$1$	1	1
$2$	11	11
$3$	111	111
$\omega$	w	12
$\omega +1$	1w	121
$\omega +2$	11w	1211
$\omega 2$	w1	1212
$\omega 2+1$	1w1	12121
$\omega 3$	w11	121212
$\omega 3+1$	1w11	1212121
${\omega }^2$	ww	122
${\omega }^2+1$	1ww	1221
${\omega }^2+\omega$	w1ww	12212
${\omega }^2+\omega +1$	1w1ww	122121
${\omega }^2+\omega 2$	w11ww	1221212
${\omega }^2+\omega 2+1$	1w11ww	12212121
${\omega }^2+\omega 3$	w111ww	122121212
${\omega }^{2}2$	ww1	122122
${\omega }^{2}2+\omega$	w1ww1	12212212
${\omega }^{2}2+\omega 2$	w11ww1	1221221212
${\omega }^{2}3$	ww11	122122122
${\omega }^3$	www	1222
${\omega }^3+\omega$	w1www	122212
${\omega }^3+{\omega }^2$	ww1www	1222122
${\omega }^3+{\omega }^2+1$	1ww1www	12221221
${\omega }^3+{\omega }^2+\omega$	w1ww1www	122212212
${\omega }^3+{\omega }^2+\omega +1$	1w1ww1www	1222122121
${\omega }^3+{\omega }^2+\omega 2$	w11ww1www	12221221212
${\omega }^3+{\omega }^{2}2+\omega$	w1ww11www	122212212212
${\omega }^3+{\omega }^{2}2+\omega 2+1$	1w11ww11www	122212212212121
${\omega }^{3}2$	www1	12221222
${\omega }^4$	wwww	12222
${\omega }^5$	wwwww	122222
${\omega }^{\omega }$	wtw	123
${\omega }^{\omega }+1$	1wtw	1231
${\omega }^{\omega }+\omega$	w1wtw	12312
${\omega }^{\omega }+\omega 2$	w11wtw	1231212
${\omega }^{\omega }+{\omega }^2$	ww1wtw	123122
${\omega }^{\omega }2$	w1tw	123123
${\omega }^{\omega }2+{\omega }^2$	ww1w1tw	123123122
${\omega }^{\omega }3$	w11tw	123123123
${\omega }^{\omega +1}$	wt1w	1232
${\omega }^{\omega +1}+{\omega }^2$	ww1wt1w	1232122
${\omega }^{\omega +1}+{\omega }^{\omega }$	wtwr1wt1w	1232123
${\omega }^{\omega +1}+{\omega }^{\omega }2$	w1twr1wt1w	1232123123
${\omega }^{\omega +1}2+{\omega }^{\omega }$	wtwr1w1t1w	12321232123
${\omega }^{\omega +1}2+{\omega }^{\omega }2$	w1twr1w1t1w	12321232123123
${\omega }^{\omega +1}3$	w11t1w	123212321232
${\omega }^{\omega +2}$	wt11w	12322
${\omega }^{\omega +3}$	wt111w	123222
${\omega }^{\omega 2}$	wtw1	12323
${\omega }^{\omega 2+1}$	wt1w1	123232
${\omega }^{\omega 2+1}+{\omega }^{\omega 2}$	wtw1r1wt1w1	12323212323
${\omega }^{\omega 2+1}+{\omega }^{\omega 2}2$	w1tw1r1wt1w1	1232321232312323
${\omega }^{\omega 2+1}2+{\omega }^{\omega 2}2$	w1tw1r1w1t1w1	1232321232321232312323
${\omega }^{\omega 2+1}2+{\omega }^{\omega 2}2+{\omega }^{\omega +1}2+{\omega }^{\omega }2$	w1twr1w1t1wr1w1tw1r1w1t1w1	123232123232123231232312321232123123
${\omega }^{\omega 2+2}$	wt11w1	1232322
${\omega }^{\omega 2+2}+{\omega }^{\omega 2+1}$	wt1w1r1wt11w1	1232322123232
${\omega }^{\omega 3}$	wtw11	1232323
${\omega }^{\omega 3+1}+{\omega }^{\omega 3}$	wtw11r1wt1w11	123232321232323
${\omega }^{\omega 4}$	wtw111	123232323
${\omega }^{{\omega }^2}$	wtww	1233
${\omega }^{{\omega }^2+\omega +1}+{\omega }^{{\omega }^2+\omega }$	wtw1wwr1wt1w1ww	1233232123323
${\omega }^{{\omega }^{2}2}$	wtww1	1233233
${\omega }^{{\omega }^3}$	wtwww	12333
${\omega }^{{\omega }^{\omega }}$	wtwtw	1234
${\omega }^{{\omega }^{\omega 2+1}3}$	wtw11t1w1	1234343234343234343
${\omega }^{{\omega }^{\omega 2+2}}$	wtwt11w1	12343433
${\omega }^{{\omega }^{\omega 3}}$	wtwtw11	12343434
${\omega }^{{\omega }^{{\omega }^2}}$	wtwtww	12344
${\omega }^{{\omega }^{{\omega }^{\omega }}}$	wtwtwtw	12345
${\epsilon }_0$	e	124
${\epsilon }_0+\omega$	w1e	12412
${\epsilon }_0+{\omega }^{\omega }$	wtwr1e	124123
${\epsilon }_{0}2$	e1	124124
${\epsilon }_{0}2+\omega$	w1e1	12412412
${\epsilon }_0\omega ={\omega }^{{\epsilon }_0+1}$	wt1e	1242
${\omega }^{{\epsilon }_0+1}+{\epsilon }_0$	e1wt1e	1242124
${\omega }^{{\epsilon }_0+1}2$	w1t1e	12421242
${\omega }^{{\epsilon }_0+2}$	wt11e	12422
${\omega }^{{\epsilon }_0+\omega }$	wtw1e	12423
${\omega }^{{\epsilon }_0+{\omega }^{\omega }}$	wtwtwr1e	124234
${\omega }^{{\epsilon }_{0}2}$	wte1	12424
${\omega }^{\omega }{}^{{\epsilon }_0+1}$	wtwt1e	1243
${\omega }^{\omega }{}^{{\epsilon }_{0}2}$	wtwte1	12435
${\epsilon }_1$	ee	1244
${\omega }^{{\epsilon }_{1}2}$	wtee1	1244244
${\epsilon }_2$	eee	12444
${\epsilon }_{\omega }$	etw	1245
${\epsilon }_{\omega }+{\omega }^{\omega }$	wtwr1etw	1245123
${\epsilon }_{\omega }+{\epsilon }_0$	e1etw	1245124
${\epsilon }_{\omega }2$	e1tw	12451245
${\omega }^{{\epsilon }_{\omega }+1}$	wt1etw	12452
${\omega }^{{\epsilon }_{\omega }+\omega }$	wtw1etw	124523
${\omega }^{{\epsilon }_{\omega }2}$	wte1tw	1245245
${\epsilon }_{\omega +1}$	et1w	12454
${\epsilon }_{\omega 2}$	etw1	124545
${\epsilon }_{{\omega }^{\omega }}$	etwtw	12456
${\epsilon }_{{\epsilon }_0}$	ete	12457
${\epsilon }_{{\epsilon }_0+1}$	et1e	12457
${\epsilon }_{{\epsilon }_1}$	etee	12457
${\epsilon }_{{\epsilon }_{{\epsilon }_0}}$	etete	124578A
${\zeta }_0$	y	1246