Why does Arya not mention Ilyn Payne in her list? [duplicate]

Question

Ser Ilyn Payne is still alive and Arya stopped mentioning him in her death list; after all he was the one who beheaded Ned Stark in the first place. So why has he fallen off her list?

Answer 1

Your question is based on an erroneous premise. Arya has not stopped mentioning Ilyn Payne in her prayer. The Hound is the only one whom she omitted while he was alive. The list kept shortening as the people on it kept dying. Let's list down all the instances of her prayers, as they appear in all books:

A Clash of Kings

Every night Arya would say their names. "Ser Gregor," she'd whisper to her stone pillow. "Dunsen, Polliver, Chiswyck, Raff the Sweetling. The Tickler and the Hound. Ser Amory, Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei." Back in Winterfell, Arya had prayed with her mother in the sept and with her father in the godswood, but there were no gods on the road to Harrenhal, and her names were the only prayer she cared to remember.
_{ACOK - Arya VI}

And

It took him only three days to earn the place of honor in her nightly prayers. "Weese," she would whisper, first of all. "Dunsen, Chiswyck, Polliver, Raff the Sweetling. The Tickler and the Hound. Ser Gregor, Ser Amory, Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei." If she let herself forget even one of them, how would she ever find him again to kill him?
_{ACOK - Arya VII}

And

"Dunsen, Chiswyck, Polliver, Raff the Sweetling. The Tickler and the Hound. Ser Gregor, Ser Amory, Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei." She thought she might add three more names to her prayer, but she was too tired to decide tonight.
_{ACOK - Arya VII}

And

"Weese," Arya whispered that night as she bent over the tear in her shift. "Dunsen, Polliver, Raff the Sweetling," she said, calling a name every time she pushed the bone needle through the undyed wool. "The Tickler and the Hound. Ser Gregor, Ser Amory, Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei." She wondered how much longer she would have to include Weese in her prayer, and drifted off to sleep dreaming that on the morrow, when she woke, he'd be dead.
_{ACOK - Arya VIII}

And

She slashed at birch leaves till the splintery point of the broken broomstick was green and sticky. "Ser Gregor," she breathed. "Dunsen, Polliver, Raff the Sweetling." She spun and leapt and balanced on the balls of her feet, darting this way and that, knocking pinecones flying. "The Tickler," she called out one time, "the Hound," the next. "Ser Ilyn, Ser Meryn, Queen Cersei." The bole of an oak loomed before her, and she lunged to drive her point through it, grunting "Joffrey, Joffrey, Joffrey."
_{ACOK - Arya X}

A Storm of Swords

Arya looked up at the fleshless dead in their wet rotting clothes and said her own prayer. Ser Gregor, it went, Dunsen, Polliver, Raff the Sweetling. The Tickler and the Hound. Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei. She ended it with valar morghulis
_{ASOS - Arya I}

Then we see that she omitted some names in the same chapter but only because she fell asleep midway:

The last thing she did before closing her eyes was unsheathe her sword and lay it down beside her. "Ser Gregor," she whispered, yawning. "Dunsen, Polliver, Raff the Sweetling. The Tickler and . . . the Tickler . . . the Hound . . ."
_{ASOS - Arya I}

Then she is back to normal:

"Valar morghulis," she would say under her breath. "Ser Gregor, Dunsen, Polliver, Raff the Sweetling. The Tickler and the Hound. Ser Ilyn, Ser Meryn, Queen Cersei, King Joffrey."
_{ASOS - Arya III}

Again

"Ser Gregor, Dunsen, Polliver, Raff the Sweetling. The Tickler and the Hound. Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei." Arya tried to imagine how they would look when they were dead, but it was hard to bring their faces to mind.
_{ASOS - Arya VII}

And

Arya huddled down and held her tongue. Valar morghulis, she thought sullenly. Ser Ilyn, Ser Meryn, King Joffrey, Queen Cersei. Dunsen, Poliver, Raff the Sweetling, Ser Gregor and the Tickler. And the Hound, the Hound, the Hound.
_{ASOS - Arya IX}

And

"Ser Gregor the Mountain," she said softly. "Dunsen, Raff the Sweetling, Ser Ilyn, Ser Meryn, Queen Cersei." It made her feel queer to leave out Polliver and the Tickler. And Joffrey too.
_{ASOS - Arya XIII}

A Feast for Crows

The Titan's eyes seemed brighter now, and farther apart. Arya did not know any Many-Faced God, but if he answered prayers, he might be the god she sought. Ser Gregor, she thought, Dunsen, Raff the Sweetling, Ser Ilyn, Ser Meryn, Queen Cersei. Only six now. Joffrey was dead, the Hound had slain Polliver, and she'd stabbed the Tickler herself, and that stupid squire with the pimple. I wouldn't have killed him if he hadn't grabbed me.
_{AFFC - Arya I}

And

Each night before sleep, she murmured her prayer into her pillow. "Ser Gregor," it went. "Dunsen, Raff the Sweetling, Ser Ilyn, Ser Meryn, Queen Cersei." She would have whispered the names of the Freys of the Crossing too, if she had known them. One day I'll know, she told herself, and then I'll kill them all.
_{AFFC - Arya II}

And also

So she prayed her own prayer to the Many-Faced God, the one that went, "Ser Gregor, Dunsen, Raff the Sweetling, Ser Ilyn, Ser Meryn, Queen Cersei." She prayed in silence. If the Many-Faced God was a proper god, he would hear her.
_{AFFC - Arya II}

And

"Ser Gregor," she chanted, as she crossed a stone bridge supported by four arches. From the center of its span she could see the masts of ships in the Ragman's Harbor. "Dunsen, Raff the Sweetling, Ser Ilyn, Ser Meryn, Queen Cersei." Rain began to fall. Arya turned her face up to let the raindrops wash her cheeks, so happy she could dance. "Valar morghulis," she said, "valar morghulis, valar morghulis."
_{AFFC - Arya II}

A Dance with Dragons

Ser Gregor, she thought. Dunsen, Raff the Sweetling. Ser Ilyn, Ser Meryn, Queen Cersei. Her morning prayer. Or was it? No, she thought, not mine. I am no one. That is the night wolf's prayer. Someday she will find them, hunt them, smell their fear, taste their blood. Someday.
_{ADWD - The Blind Girl}

And

Ser Gregor, she could not help but think. Dunsen, Raff the Sweetling. Ser Ilyn, Ser Meryn, Queen Cersei. If she spoke, she would need to lie, and he would know. She kept silent.
_{ADWD - The ugly little Girl}

And

Ser Gregor, Dunsen, Raff the Sweetling. Ser Ilyn, Ser Meryn, Queen Cersei. She mouthed the names silently. In the House of Black and White, you never knew who might be listening.
_{ADWD - The ugly little Girl}

Conclusion

As you can see, not even once has Arya failed to mention Ilyn Payne except that one time in ASOS when she fell asleep.

Answer 2

As @Aegon's answer tells us that there is confusion in the question, I would like to explain where this confusion is coming from as remembering the differences between the show and the books can be quite exhausting.

From the Game of Thrones perspective Ser Ilyn Payne is (presumably) still alive. Although we have not seen him since S02E09 "Blackwater", he was mentioned in S04E10 by Tywin in conversation with Tyrion.

As to why we have not seen him... it was because the actor, Wilko Johnson, was diagnosed with terminal cancer in 2013. He retired from acting and announced a farewell tour (Johnson was a musician before acting). Although he did get better in 2015, it seems that the writers simply let his character fall into obscurity rather than giving him a more ultimate send off.

Considering we are entering the 8th and final season of the show and the character has not been seen or heard of in three seasons it safe to assume that that he will not be reappearing. So from an in-universe perspective Arya probably believes that Ser Ilyn is dead.