A380195 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 under-under-down dealing is used, then the resulting cards will be dealt in increasing order.

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

Offset: 1

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Jan 15 2025

Keywords

• nonn
• tabl

Comments

Under-under-down dealing is a dealing pattern where the top card is placed at the bottom of the deck, then the next card is also placed at the bottom of the deck, and then the next 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 two people are 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 A381667 is an inverse permutation of the corresponding row of this triangle.

The total number of moves for row n is 3n.

The first column is A381591, the order of elimination of the first person in the Josephus problem.

The index of the largest number in row n is A054995(n), corresponding to the index of the freed person in the corresponding Josephus problem.

T(n,3j) = j, for 3j <= n.

Examples

			Consider a deck of four cards arranged in the order 4,2,1,3. Card 4 goes under, card 2 goes under, and card 1 is dealt. Now the deck is ordered 3,4,2. Card 3 goes under, card 4 goes under, and card 2 is dealt. Now the leftover deck is ordered 3,4. Card 3 goes under, card 4 goes under, and card 3 is dealt. Then card 4 is dealt. The dealt cards are in order. Thus, the fourth row of the triangle is 4,2,1,3.
Table begins:
  1 ;
  1, 2;
  2, 3, 1;
  4, 2, 1, 3;
  2, 4, 1, 5, 3;
  6, 4, 1, 3, 5, 2;
  6, 3, 1, 7, 5, 2, 4;
  3, 5, 1, 7, 4, 2, 8, 6;
  9, 7, 1, 4, 6, 2, 8, 5, 3;
		

Crossrefs

Cf. A006257, A054995, A225381, A321298, A378635, A380195, A008585, A381591 A381667.

Programs

Python

def UUD(n):
    return invPerm(UUDES(n))
def UUDES(n):
    l=[]
    for i in range(n):
        l.append(i+1)
    index = 0
    P=[]
    for i in range(n):
        index+=2
        index=index%len(l)
        P.append(l[index])
        l.pop(index)
    return P
def invPerm(p):
    inv = []
    for i in range(len(p)):
        inv.append(None)
    for i in range(len(p)):
        inv[p[i]-1]=i+1
    return inv
sequence = []
for i in range(1,20):
    sequence += [str(v) for v in UUD(i)]
print(", ".join(sequence))

Python

def row(n):
    i, J, out = 0, list(range(1, n+1)), []
    while len(J) > 0:
        i = (i + 2)%len(J)
        out.append(J.pop(i))
    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, Mar 27 2025

Formula

T(n, 3 mod n) = 1

T(n, k) = T(n-1, k-3 mod n) + 1 if k != 3 mod n, where "a mod b" denotes the representative of the a's modular equivalence class in {0, ..., b-1}.