A381612 Irregular triangle read by rows: T(n,k) for n-1 <= k <= A328378(n) is the number of permutations of length n having a sum of differences equal to k.
2, 2, 4, 2, 4, 12, 4, 2, 2, 4, 14, 32, 18, 28, 14, 8, 2, 4, 16, 36, 92, 68, 128, 92, 122, 72, 64, 16, 8, 2, 4, 18, 40, 112, 240, 256, 448, 438, 668, 502, 696, 480, 496, 264, 240, 88, 48, 2, 4, 20, 44, 134, 288, 696, 776, 1566, 1620, 2788, 2524, 3914, 3192, 4544, 3376, 4056, 2720, 2960, 1776, 1712, 816, 576, 144, 72
Offset: 2
Examples
Triangle begins:
n: row n
---------------
2: 2;
3: 2, 4;
4: 2, 4, 12, 4, 2;
5: 2, 4, 14, 32, 18, 28, 14, 8;
6: 2, 4, 16, 36, 92, 68, 128, 92, 122, 72, 64, 16, 8;
7: 2, 4, 18, 40, 112, 240, 256, 448, 438, 668, 502, 696, 480, 496, 264, 240, 88, 48;
...
First item in each row corresponds to sum at n-1, second item to sum at n, etc.
Row 4:
|1, 2, 3, 4|=3, |1, 3, 2, 4|=5, |3, 1, 2, 4|=5, |1, 4, 2, 3|=6,
|4, 3, 2, 1|=3, |1, 3, 4, 2|=5, |3, 2, 1, 4|=5, |2, 3, 1, 4|=6,
|1, 2, 4, 3|=4, |1, 4, 3, 2|=5, |3, 4, 1, 2|=5, |3, 2, 4, 1|=6,
|2, 1, 3, 4|=4, |2, 1, 4, 3|=5, |4, 1, 2, 3|=5, |4, 1, 3, 2|=6,
|3, 4, 2, 1|=4, |2, 3, 4, 1|=5, |4, 2, 1, 3|=5, |2, 4, 1, 3|=7,
|4, 3, 1, 2|=4, |2, 4, 3, 1|=5, |4, 2, 3, 1|=5, |3, 1, 4, 2|=7,
Thus, for example, T(4,5)=12 because there are 12 permutations with a sum of differences equal to 5.
Links
- Norman Whitehead, Log10 Graph of Several Rows
Programs
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Python
from itertools import permutations def count_l1_sums(n): nm1 = n-1 a000982 = (((nm1*nm1)+1)//2) # from OEIS tally = a000982*[0] # number of different totals perms = permutations([i for i in range(n)]) # length is factorial(n) # Compute each permutation's first difference for perm in perms: l1_norm = 0 for i in range(nm1): diff = perm[i+1]-perm[i] if diff>0: l1_norm += diff else: l1_norm -= diff tally[l1_norm-nm1] += 1 # diff==(n-1) goes into first position return tally
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