A386639 - OEIS

A386639

Triangle T(n,k) read by rows, where row n is a permutation of the numbers 1 through n, such that if a deck of n cards is prepared in this order, and the AP dealing is used, then the resulting cards will be dealt in increasing order.

1, 2, 1, 3, 1, 2, 2, 1, 4, 3, 3, 1, 4, 5, 2, 4, 1, 6, 3, 2, 5, 5, 1, 3, 4, 2, 6, 7, 3, 1, 7, 5, 2, 6, 8, 4, 7, 1, 8, 6, 2, 9, 4, 5, 3, 9, 1, 8, 5, 2, 4, 7, 6, 3, 10, 5, 1, 6, 4, 2, 10, 11, 7, 3, 8, 9, 7, 1, 4, 9, 2, 11, 10, 8, 3, 6, 5, 12, 4, 1, 13, 11, 2, 10, 6, 7, 3, 5, 12, 9, 8, 10, 1, 7, 6, 2, 12, 8, 5, 3, 14

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OFFSET

1,2

COMMENTS

The AP dealing is a dealing pattern where x cards are placed at the bottom of the deck, and then the next card is dealt. The number of cards x placed at the bottom changes with every dealt card according to the arithmetic progression 1, 2, 3, and so on. This pattern repeats until all of the cards have been dealt.

This card dealing can equivalently be seen as a variation on the Josephus problem, where one person is skipped, then the next person is eliminated, then two people are skipped and one person is eliminated, then three people are skipped, and so on. T(n,k) is the order of elimination of the k-th person in the Josephus problem. Equivalently, each row of T is the inverse permutation of the corresponding row of the Josephus triangle A386641, i.e., A386641(n,T(n,k)) = k.

The total number of moves for row n is A000096.

The first column is A386643(n), the order of elimination of the first person in the Josephus problem.

The index of the largest number in row n is A291317(n), corresponding to the index of the freed person in the corresponding Josephus problem.

LINKS

Table of n, a(n) for n=1..101.

Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See pp. 17, 39.

FORMULA

T(n,A000096(k)) = k, for A000096(k) <= n.

EXAMPLE

Consider a deck of four cards arranged in the order 2,1,4,3. We put one card under and deal the next card, which is card number 1. Now the deck is ordered 4,3,2. We place 2 cards under and deal the next one, which is card number 2. Now the deck is 4,3. Again, placing 3 cards under and dealing the next, we will deal card number 3, leaving card number 4 to be dealt last. The dealt cards are in order. Thus, the fourth row of the triangle is 2,1,4,3.

The triangle begins as follows:

2, 1;

3, 1, 2;

2, 1, 4, 3;

3, 1, 4, 5, 2;

4, 1, 6, 3, 2, 5;

5, 1, 3, 4, 2, 6, 7;

3, 1, 7, 5, 2, 6, 8, 4;

7, 1, 8, 6, 2, 9, 4, 5, 3;

CROSSREFS

Cf. A000096, A291317, A386641, A386643.

Cf. A378635 (classical elimination process).