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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.

%I A092113 #25 Dec 24 2023 01:10:19
%S A092113 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,
%T A092113 30,149,543,1357,1903,1016,35,1,7,42,251,1191,4281,10561,15011,8520,
%U A092113 455,1,8,56,391,2278,10666,38015,93585,132697,79379,5804,1,9,72,575
%N A092113 Triangle read by rows: T(n,k) is the number of stacks of n pancakes requiring k = 0, ..., A058986(n) flips to sort.
%C A092113 Last term of row k is A067607(k).
%C A092113 Row n has length A058986(n) + 1. - _Martin Renner_, Jul 23 2017
%H A092113 Sean A. Irvine, <a href="/A092113/b092113.txt">Table of n, a(n) for n = 1..97</a> (terms 1..68 from Martin Renner)
%e A092113 Triangle begins:
%e A092113   1;
%e A092113   1,  1;
%e A092113   1,  2,  2,  1;
%e A092113   1,  3,  6, 11,  3;
%e A092113   1,  4, 12, 35, 48, 20;
%e A092113   ...
%e A092113 From _Jon E. Schoenfield_, Dec 16 2021: (Start)
%e A092113 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,
%e A092113 T(3,0)=1 permutation (namely, {1,2,3}) requires no prefix reversals (because it is already sorted);
%e A092113 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};
%e A092113 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
%e A092113 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};
%e A092113 thus, the terms in row n=3 are 1, 2, 2, 1. (End)
%Y A092113 Cf. A058986, A067607.
%K A092113 nonn,tabf
%O A092113 1,5
%A A092113 _Eric W. Weisstein_, Feb 21 2004