Question
How many permutations of the YHVH Name are there ?
Answer
This topic is very deep, it is widely discussed in the Talmud, in the Rambam and in Kabala.
The name YHVH itself of four letters appears in the Torah vowelized sometimes like the name of Adnut and sometimes like the name of Elokut, according to the context.
The Gemara in Kiddushin 71. mentions a name of 12 letters and a name of 42 letters that are based on the Tetragrammaton YHVH.
The Radbaz quotes Sefer Chayey Olam that the name of 72 letters is the Tetragrammaton.
In the commentary to the prayer for livelihood that is in Birkat Cohanim by my great grandfather Rabbi Shmuel Deutchlander Z”L, he brings the Midrash Yalkut Reuveni [also brought in Rabeinu Bechayey Parashat Vayeshev] “and he made him a coat of many colors” teaches us that he handed down to him the secret of 22 letters, and Rabeinu Bechayey explains that the meaning is that he “dressed” him with the knowledge of wisdom of the 22 letter name like he learned from Shem and Eiver.
The Radbaz quotes Sefer Hakabala that the name of 22 letters comes out of Birkat Cohanim.
The topic is discussed in detail in Tikunei Hazohar, all 72 names that are based on the name YHVH are spelled out in the introduction page 8:
For more information see, the Rambam in Moreh Nevuchim, Shut HaRadbaz Shut Chatam Sofer, Sefer Shemot Kodesh Ve’Chol, Rabinu Bechayey and the Chatam Sofer on the Torah Parashat Vayeshev, Commentary to the prayer for livelihood that is in Birkat Cohanim and Sefer Melechet Hakodesh.
[קידושין עא. תיקוני הזוהר הקדמה דף ח: מורה נבוכים להרמב"ם חלק א פרק סא וסב, ספר שמות קודש וחול המיוחס לרמב"ם עם פירוש נזר הקודש לרב יצחק רצאבי עמ' יא-יז, שו"ת הרדב"ז ח"ה ללשונות הרמב"ם סימן לד, שו"ת חת"ס חו"מ סימן קצב, רבנו בחיי וחת"ס עה"ת פרשת וישב עה"פ ועשה לו כתונת פסים, פירוש התפילה על הפרנסה אשר בברכת כהנים לזקני הגאון רבי שמואל דייטשלענדער זצ"ל בעהמח"ס מראה הלבנון רב בק"ק פוחאף, ספר מלאכת הקדש לרבי אלעזר פלעקלס מפראג].
