A381051 -id:A381051 - cp's OEIS frontend

A381050 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 down-under-down dealing is used, then the resulting cards will be dealt in increasing order.

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

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Apr 14 2025

Down-under-down dealing is a dealing pattern where the top card is dealt, the second card is placed at the bottom of the deck, then the third card is dealt. This pattern repeats until all of the cards have been dealt.
This card dealing is related to a variation on the Josephus problem, where the first person is eliminated, the second person is skipped, and the third person is eliminated. The card in row n and column k is x if and only if in the corresponding Josephus problem with n people, the person number x is the k-th person eliminated. Equivalently, each row of Josephus triangle A383076 is an inverse permutation of the corresponding row of this triangle.
The total number of moves for row n is A032766(n) = floor(3n/2).
The index of the largest number in row n is A381051(n), corresponding to the index of the freed person in the corresponding Josephus problem.

			Consider a deck of four cards arranged in the order 1,4,2,3. In round 1, card 1 is dealt, card 4 goes under, card 2 is dealt. Now the deck is ordered 3,4. In round 2, card 3 is dealt, card 4 goes under, then card 4 is dealt. The dealt cards are in order. Thus, the fourth row of the triangle is 1,4,2,3.
Table begins:
  1;
  1, 2;
  1, 3, 2;
  1, 4, 2, 3;
  1, 4, 2, 3, 5;
  1, 5, 2, 3, 6, 4;
  1, 7, 2, 3, 6, 4, 5;
  1, 6, 2, 3, 7, 4, 5, 8;

Cf. A006257, A032766, A225381, A321298, A378635, A381050, A381051, A383076, A378674.

row[n_]:=Module[{ds,res,k,i=1,len},ds=CreateDataStructure["Queue",Range[n]];res=CreateDataStructure["FixedArray",n];While[(ds["Length"]>=2),res["SetPart",i++,ds["Pop"]];ds["Push",ds["Pop"]];If[ds["Length"]>1,res["SetPart",i++,ds["Pop"]];]];res["SetPart",n,ds["Pop"]];Flatten[PositionIndex[res["Elements"]]/@Range[n]]];
Array[row, 13, 1] // Flatten (* Shenghui Yang, May 11 2025 *)

def row(n):
    i, J, out = 0, list(range(1, n+1)), []
    while len(J) > 1:
        i = i%len(J)
        out.append(J.pop(i))
        i = (i + 1)%len(J)
        #i = i%len(J)
        if len(J) > 1:
            out.append(J.pop(i))
    out += [J[0]]
    return [out.index(j)+1 for j in list(range(1, n+1))]
print([e for n in range(1, 14) for e in row(n)]) # Michael S. Branicky, Apr 28 2025

T(n,3j) = 2j, for 3j <= n. T(n,3j+1) = 2j+1, for 3j+1 <= n.

A383846 A version of the Josephus problem: a(n) is the surviving integer under the eliminate-eliminate-skip version of the elimination process.

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 6, 3, 9, 6, 3, 9, 6, 12, 9, 15, 12, 18, 15, 3, 18, 6, 21, 9, 24, 12, 27, 15, 3, 18, 6, 21, 9, 24, 12, 27, 15, 30, 18, 33, 21, 36, 24, 39, 27, 42, 30, 45, 33, 48, 36, 51, 39, 54, 42, 3, 45, 6, 48, 9, 51, 12, 54, 15, 57, 18, 60, 21, 63, 24, 66, 27

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, May 12 2025

nonn

This variation of the Josephus problem is related to down-down-under card dealing.

Cf. A006257, A383845, A383846, A001651, A383847, A337191, A381051.

Consider 4 people in a circle in order 1,2,3,4. In the first round, person 1 is eliminated, then person 2 is eliminated, then person 3 is skipped. Now people are in order 4,3. In the second round, person 4 is eliminated. The last person, person 3, is freed. Thus, a(4) = 3.

def a(n):
    i, J, out = 0, list(range(1, n+1)), []
    while len(J) > 1:
        J.pop(i)
        i = i%len(J)
        if len(J) > 1:
            J.pop(i)
        i = i%len(J)
        i = (i + 1)%len(J)
    return J[0]
print([a(n) for n in range(1, 73)])

A383076 Triangle T(n,k) read by rows: where T(n,k) is the number of the k-th eliminated person in the variation of the Josephus elimination process for n people, where the elimination pattern is eliminate-skip-eliminate.

Original entry on oeis.org

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

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Apr 15 2025

This Josephus problem is related to down-under-down card dealing.
The n-th row has n elements.
In this variation of the Josephus elimination process, the numbers 1 through n are arranged in a circle. A pointer starts at position 1. With each turn, the pointer eliminates the first number, skips the second, then eliminates the third. The process repeats until no numbers remain. This sequence represents the triangle T(n, k), where n is the number of people in the circle, and T(n, k) is the elimination order of the k-th number in the circle.

			Consider 4 people in a circle. Initially, person number 1 is eliminated, person number 2 is skipped, and person number 3 is eliminated. The remaining people are now in order 4, 2. Then, 4 is eliminated, and 2 is left. Thus, the fourth row of the triangle is 1, 3, 4, 2, the order of elimination.
Triangle begins;
1;
1, 2;
1, 3, 2;
1, 3, 4, 2;
1, 3, 4, 2, 5;
1, 3, 4, 6, 2, 5;
1, 3, 4, 6, 7, 5, 2;
1, 3, 4, 6, 7, 2, 5, 8;

Cf. A006257, A032766, A225381, A321298, A378635, A381050, A381051.

def row(n):
    i, J, out = 0, list(range(1, n+1)), []
    while len(J) > 1:
        i = i%len(J)
        out.append(J.pop(i))
        i = (i + 1)%len(J)
        if len(J) > 1:
            out.append(J.pop(i))
    out += [J[0]]
    return out
print([e for n in range(1, 14) for e in row(n)]) # Michael S. Branicky, Apr 28 2025

cp's OEIS Frontend

A381050 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 down-under-down dealing is used, then the resulting cards will be dealt in increasing order.

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A383846 A version of the Josephus problem: a(n) is the surviving integer under the eliminate-eliminate-skip version of the elimination process.

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A383076 Triangle T(n,k) read by rows: where T(n,k) is the number of the k-th eliminated person in the variation of the Josephus elimination process for n people, where the elimination pattern is eliminate-skip-eliminate.

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Python