A322069 -id:A322069 - cp's OEIS frontend

A322070 Number of permutations f of {1,...,n} with f(1) < f(n) such that Sum_{k=1..n-1} 1/(f(k)+f(k+1)) = 1.

0, 0, 0, 0, 0, 0, 1, 8, 22, 98, 844, 3831, 20922, 88902, 358253

Offset: 1

Zhi-Wei Sun, Nov 25 2018

Conjecture 1: a(n) > 0 for all n > 6. In other words, for each n = 7,8,... we have Sum_{k=1..n-1} 1/(f(k)+f(k+1)) = 1 for some permutation f in the symmetric group S_n.
Conjecture 2: For any integer n > 7, there is a permutation g of {1,...,n} such that 1/(g(1)+g(2)) + 1/(g(2)+g(3)) + ... + 1/(g(n-1)+g(n)) + 1/(g(n)+g(1)) = 1.
Conjecture 3. For any integer n > 5, there is a permutation p of {1,...,n} such that Sum_{k=1..n-1} 1/(p(k)-p(k+1)) = 0.
Conjecture 4. For each integer n > 7, there is a permutation q of {1,...,n} such that 1/(q(1)-q(2)) + 1/(q(2)-q(3)) + ... + 1/(q(n-1)-q(n)) + 1/(q(n)-q(1)) = 0.
We have verified all the four conjectures for n up to 11. For Conjecture 2 with n = 8, we may take (g(1),...,g(8)) = (1,3,5,4,6,2,7,8) since 1/(1+3) + 1/(3+5) + 1/(5+4) + 1/(4+6) + 1/(6+2) + 1/(2+7) + 1/(7+8) + 1/(8+1) = 1.
See also A322069 for a similar conjecture.

			a(7) = 1, and for the permutation (4,5,7,2,1,3,6) of {1,...,7} we have 1/(4+5) + 1/(5+7) + 1/(7+2) + 1/(2+1) + 1/(1+3) + 1/(3+6) = 1.

Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.

Cf. A322069.

V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]];
Do[r=0;Do[If[Part[V[n],k][[1]]>=Part[V[n],k][[n]]||Sum[1/(Part[V[n],k][[i]]+Part[V[n],k][[i+1]]),{i,1,n-1}]!=1,Goto[aa]];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]

a(12)-a(15) from Hugo Pfoertner, Aug 20 2022

A322099 Number of permutations f of {1,...,n} with f(1) < f(n) and Sum_{k=1..n-1} 1/(f(k)^2 - f(k+1)^2) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 4, 55, 78, 552, 3484, 12720

Offset: 1

Zhi-Wei Sun, Nov 26 2018

Conjecture: a(n) > 0 for all n > 7. In other words, for each n = 8,9,... we have Sum_{k=1..n-1} 1/(f(k)^2 - f(k+1)^2) = 0 for some permutation f in the symmetric group S_n.

			a(8) = 1, and for the permutation (4,5,2,7,3,1,6,8) of {1,...,8} we have 1/(4^2-5^2) + 1/(5^2-2^2) + 1/(2^2-7^2) + 1/(7^2-3^2) + 1/(3^2-1^2) + 1/(1^2-6^2) + 1/(6^2-8^2) = 0.
a(12) > 0 since for the permutation (1,3,7,5,4,8,6,2,10,11,9,12) of {1,...,12} we have 1/(1^2-3^2) + 1/(3^2-7^2) + 1/(7^2-5^2) + 1/(5^2-4^2) + 1/(4^2-8^2) + 1/(8^2-6^2) + 1/(6^2-2^2) + 1/(2^2-10^2) + 1/(10^2-11^2) + 1/(11^2-9^2) + 1/(9^2-12^2) = 0.
a(13) > 0 since for the permutation (1,6,2,9,11,5,3,7,13,8,4,10,12) of {1,...,13} we have 1/(1^2-6^2) + 1/(6^2-2^2) + 1/(2^2-9^2) + 1/(9^2-11^2) + 1/(11^2-5^2) + 1/(5^2-3^2) + 1/(3^2-7^2) + 1/(7^2-13^2) + 1/(13^2-8^2) + 1/(8^2-4^2) + 1/(4^2-10^2) + 1/(10^2-12^2) = 0.

Zhi-Wei Sun, On restricted permutations of {1,...,n}, arXiv:1811.10503 [math.CO], 2018.
Zhi-Wei Sun, Is there a permutation pi in S_n with Sum_{k=1..n-1} 1/(pi(k)^2-pi(k+1)^2) = 0 for each n > 7?, Question 316208 on Mathoverflow, Nov. 25, 2018.

Cf. A322069, A322070, A356446.

V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]];
Do[r=0;Do[If[Part[V[n],k][[1]]>=Part[V[n],k][[n]]||Sum[1/(Part[V[n],k][[i]]^2-Part[V[n],k][[i+1]]^2),{i,1,n-1}]!=0,Goto[aa]];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]

a(n)={my(s=0); forperm(n, f, if(f[1]Andrew Howroyd, Nov 27 2018

a(12) from Andrew Howroyd, Nov 27 2018

a(13)-a(15) from Hugo Pfoertner, Aug 22 2022

A356446 Number of permutations f of {1,...,n} with f(1) = 2 and f(2) = 1 such that the numbers f(k)f(k+1) (0 < k < n) are distinct and Sum_{k=1..n-1} 1/(f(k)f(k+1)) = 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 11, 7, 61, 388, 2933, 2741

Offset: 2

Zhi-Wei Sun, Aug 07 2022

Conjecture: a(n) > 0 for all n > 5.

This is stronger than Conjecture 1 in A322069.

			a(6) = 1 with the permutation (f(1),...,f(6)) = (2,1,3,4,5,6) meeting the requirement. Note that 2*1 = 2, 1*3 = 3, 3*4 = 12, 4*5 = 6 and 5*6 = 30 are pairwise distinct and 1/2 + 1/3 + 1/12 + 1/20 + 1/30 = 1.
a(9) = 1 with the permutation (f(1),...,f(9)) = (2,1,3,8,7,6,5,9,4) meeting the requirement. Note that 2*1 = 2, 1*3 = 3, 3*8 = 24, 8*7 = 56, 7*6 = 42, 6*5 = 30, 5*9 = 45 and 9*4 = 36 are pairwise distinct and 1/2 + 1/3 + 1/24 + 1/56 + 1/42 + 1/30 + 1/45 + 1/36 = 1.
a(11) = 1 with the permutation (f(1),...,f(11)) = (2,1,4,3,5,11,6,7,8,9,10) meeting the requirement. Note that 2*1 = 2, 1*4 = 4, 4*3 = 12, 3*5 = 15, 5*11 = 55, 11*6 = 66, 6*7 = 42, 7*8 = 56, 8*9 = 72 and 9*10 = 90 are pairwise distinct and 1/2 + 1/4 + 1/12 + 1/15 + 1/55 + 1/66 + 1/42 + 1/56 + 1/72 + 1/90 = 1.
a(12) > 0 since for the permutation (f(1),...,f(12)) = (2,1,4,5,3,6,8,10,11,12,7,9) the 11 numbers 2*1 = 2, 1*4 = 4, 4*5 = 20, 5*3 = 15, 3*6 = 18, 6*8 = 48, 8*10 = 80, 10*11 = 110, 11*12 = 132, 12*7 = 84 and 7*9 = 63 are pairwise distinct and 1/2 + 1/4 + 1/20 + 1/15 + 1/18 + 1/48 + 1/80 + 1/110 + 1/132 + 1/84 + 1/63 = 1. Note that f(3) = 4, f(4) = 3 and f(5) is among 5, 6, 8 for all the other permutations f of {1,...,12} meeting the requirement.

Guoniu Han, On the existence of permutations conditioned by certain rational functions, Electron. Res. Arch., 28 (2020), 149-156.
Zhi-Wei Sun, On permutations of {1,...,n} and related topics, J. Algebraic Combin., 2021.

Cf. A000961, A322069, A322070, A356187.

(* A program to find all the permutations f of {1, ..., 10} with f(1) = 2 and f(2) = 1 such that those f(k)*f(k+1) (k = 1..9) are pairwise distinct and Sum_{k=1..9} 1/(f(k)*f(k+1)) = 1. *)
V[i_]:=V[i]=Part[Permutations[{3, 4, 5, 6, 7, 8, 9, 10}], i]
m=0; Do[U={2}; Do[U=Append[U, V[i][[j]]*If[j==1, 1, V[i][[j-1]]]], {j, 1, 8}];
If[Length[Union[U]]==9&&Sum[1/U[[j]], {j, 1, 9}]==1, m=m+1; Print[m, " ", V[i], " ", U]]; Label[aa], {i, 1, 8!}]

a(13)-a(15) from Jinyuan Wang, Aug 09 2022

a(16)-a(17) from Hugo Pfoertner, Aug 19 2022

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A322070 Number of permutations f of {1,...,n} with f(1) < f(n) such that Sum_{k=1..n-1} 1/(f(k)+f(k+1)) = 1.

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A322099 Number of permutations f of {1,...,n} with f(1) < f(n) and Sum_{k=1..n-1} 1/(f(k)^2 - f(k+1)^2) = 0.

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A356446 Number of permutations f of {1,...,n} with f(1) = 2 and f(2) = 1 such that the numbers f(k)f(k+1) (0 < k < n) are distinct and Sum_{k=1..n-1} 1/(f(k)f(k+1)) = 1.

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cp's OEIS Frontend

A322070 Number of permutations f of {1,...,n} with f(1) < f(n) such that Sum_{k=1..n-1} 1/(f(k)+f(k+1)) = 1.

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A322099 Number of permutations f of {1,...,n} with f(1) < f(n) and Sum_{k=1..n-1} 1/(f(k)^2 - f(k+1)^2) = 0.

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A356446 Number of permutations f of {1,...,n} with f(1) = 2 and f(2) = 1 such that the numbers f(k)*f(k+1) (0 < k < n) are distinct and Sum_{k=1..n-1} 1/(f(k)*f(k+1)) = 1.

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A356446 Number of permutations f of {1,...,n} with f(1) = 2 and f(2) = 1 such that the numbers f(k)f(k+1) (0 < k < n) are distinct and Sum_{k=1..n-1} 1/(f(k)f(k+1)) = 1.