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A092113
Triangle read by rows: T(n,k) is the number of stacks of n pancakes requiring k = 0, ..., A058986(n) flips to sort.
3
1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 11, 3, 1, 4, 12, 35, 48, 20, 1, 5, 20, 79, 199, 281, 133, 2, 1, 6, 30, 149, 543, 1357, 1903, 1016, 35, 1, 7, 42, 251, 1191, 4281, 10561, 15011, 8520, 455, 1, 8, 56, 391, 2278, 10666, 38015, 93585, 132697, 79379, 5804, 1, 9, 72, 575
OFFSET
1,5
COMMENTS
Last term of row k is A067607(k).
Row n has length A058986(n) + 1. - Martin Renner, Jul 23 2017
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..97 (terms 1..68 from Martin Renner)
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 11, 3;
1, 4, 12, 35, 48, 20;
...
From Jon E. Schoenfield, Dec 16 2021: (Start)
For n=3, the 3! = 6 permutations are {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. Of these,
T(3,0)=1 permutation (namely, {1,2,3}) requires no prefix reversals (because it is already sorted);
T(3,1)=2 permutations (namely, {2,1,3} and {3,2,1}) require one prefix reversal, e.g., {2,1,3} -> {1,2,3};
T(3,2)=2 permutations (namely, {2,3,1} and {3,1,2}) require two prefix reversals, e.g., {2,3,1} -> {3,2,1} -> {1,2,3}; and
T(3,3)=1 permutation (namely, {1,3,2}) requires 3 prefix reversals: {1,3,2} -> {3,1,2} -> {2,1,3} -> {1,2,3};
thus, the terms in row n=3 are 1, 2, 2, 1. (End)
CROSSREFS
Sequence in context: A122888 A371830 A266378 * A331485 A342061 A045995
KEYWORD
nonn,tabf
AUTHOR
Eric W. Weisstein, Feb 21 2004
STATUS
approved