A327845 -id:A327845 - cp's OEIS frontend

A327844 Table read by antidiagonals: the m-th row gives the sequence constructed by repeatedly choosing the smallest positive number not already in the row such that for each k = 1, ..., m, the k-th differences are distinct.

1, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 4, 3, 6, 1, 2, 4, 3, 6, 10, 1, 2, 4, 3, 6, 11, 5, 1, 2, 4, 3, 6, 11, 5, 11, 1, 2, 4, 3, 6, 11, 5, 9, 7, 1, 2, 4, 3, 6, 11, 5, 9, 7, 12, 1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 9, 1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 10, 16, 1, 2, 4, 3

Offset: 1

Peter Kagey, Sep 29 2019

First row is A175498. Main diagonal is A327743.
The index of where the m-th row first differs from A327743 is 6, 15, 15, 16, 16, 194, 301, 301, 1036, 1036, 1036, 1037, ...
For example, T(6, 194) != A327743(194), but T(6, n) = A327743(n) for n < 194.

			Table begins:
1, 2, 4, 3, 6, 10, 5, 11, 7, 12,  9, 16, 8, 17, 15, 23, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 25, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 25, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 12, ...
1, 2, 4, 3, 6, 11, 5,  9, 7, 13, 10, 18, 8, 15, 27, 14, ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A175498, A327743, A327845.

A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.

Original entry on oeis.org

0, 2, 12, 120, 1320, 17856, 273056, 4772624, 92626944, 1986317024, 46556867456, 1184827221584, 32524270418432, 958020105786536

Offset: 0

Peter Kagey, Nov 22 2019

a(n) <= ((n+1)! - 2*A131502(n))*A130783(n).

Every term is even because the n-th difference of a permutation and its reversal are the same up to sign.

			For n = 2, the second differences of the (2+1)! = 6 permutations of {0,1,2} are:
[0,1,2] ->  [1, 1] ->  0,
[0,2,1] ->  [2,-1] -> -3,
[1,0,2] -> [-1, 2] ->  3,
[1,2,0] ->  [1,-2] -> -3,
[2,0,1] -> [-2, 1] ->  3, and
[2,1,0] -> [-1,-1] ->  0.
The sum of the absolute values of these second differences is 0 + 3 + 3 + 3 + 3 + 0 = 12.

Cf. A130783, A131502, A327845.

a[n_] := Block[{x, k}, k = CoefficientList[(x - 1)^n, x]; Sum[Abs[k.p], {p, Permutations@ Range[0, n]}]]; Array[a, 10, 0] (* Giovanni Resta, Nov 23 2019 *)

from math import comb
from itertools import permutations
def A329851(n):
    c = [-comb(n,i) if i&1 else comb(n,i) for i in range(n+1)]
    return sum(abs(sum(c[i]*p[i] for i in range(n+1))) for p in permutations(range(n+1)) if p[0]Chai Wah Wu, Jun 04 2024

a(10) from Alois P. Heinz, Nov 22 2019

a(11)-a(13) from Giovanni Resta, Nov 23 2019

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A327844 Table read by antidiagonals: the m-th row gives the sequence constructed by repeatedly choosing the smallest positive number not already in the row such that for each k = 1, ..., m, the k-th differences are distinct.

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A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.

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