A014138 -id:A014138 - cp's OEIS frontend

A074080 Triangle T(n,k) (listed in order T(1,0), T(2,0), T(2,1), T(3,0), T(3,1), T(3,2), T(4,0), etc.) giving the number of 2^k-cycles that occur in the n-th sub-permutation of A069767/A069768 (i.e., in the range [A014137(n-1)..A014138(n-1)] inclusive).

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 3, 5, 3, 1, 1, 0, 3, 10, 9, 4, 1, 0, 1, 3, 17, 24, 14, 5, 1, 0, 1, 3, 28, 57, 44, 20, 6, 1, 0, 0, 5, 41, 128, 128, 71, 27, 7, 1, 0, 1, 4, 60, 271, 354, 234, 106, 35, 8, 1, 0, 0, 5, 81, 549, 937, 738, 384, 150, 44, 9, 1, 0, 0, 5, 106, 1061

Offset: 0

Antti Karttunen, Aug 19 2002

			If we take the fifth such sub-permutation, i.e., the subsequence A069767[23..64]: [45,46,48,49,50,54,55,57,58,59,61,62,63,64,44,47,53,56,60,43,52,40,31,32,41,34,35,36,42,51,39,30,33,38,29,26,27,37,28,25,24,23], subtract 22 from each term and convert the resulting permutation of [1..42] to disjoint cycle notation, we get:
(17,31),(20,21,30,29),(3,26,12,40),(6,32,8,35,7,33,11,39),(15,22,18,34,16,25,19,38),(1,23,9,36,4,27,13,41,2,24,10,37,5,28,14,42)
which implies that T(5,0) = 0 (no fixed elements), T(5,1) = 1 (one transposition), T(5,2) = 2 (two 4-cycles), T(5,3) = 2 (two 8-cycles), T(5,4) = 1 (and one 16-cycle). It is guaranteed that only cycles whose length is a power of 2 occur in A069767/A069768.

Upper triangular region of the square array A074079 (actually, only the area above its main diagonal, excluding also the leftmost column). T(n, k) = A073430(n, k)/(2^k) [with the rightmost edge of A073430 discarded]. Row sums: A073431. A000108(n) = Sum_{i=0..n-1} 2^i * T(n, i). Cf. A073346, A003056, A002262.

A074079bi := (n,k) -> A073346bi(n,k)/(2^k);
A074080 := n -> A074079bi(A003056(n)+1, A002262(n));
A003056 := n -> floor(sqrt(2*(1+n))-(1/2));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

Original entry on oeis.org

1, 1, 0, 3, 0, 9, 0, 21, 0, 45, 0, 99, 0, 195, 0, 399, 0, 801, 0

Offset: 0

Antti Karttunen, Jan 27 2003

R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, C-program for counting the initial terms of this sequence
Antti Karttunen, Illustration of initial terms for trees of sizes n=3..11

Cf. A057505, A071663, A079438, A079441, A079443, A079444.

Occurs in A073202 as row 176609070820803.

For all n >= 0, a(2n+3)/3 = A079444(n).

Name corrected by Antti Karttunen, Dec 13 2017

A081150 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 18, 37, 95, 212, 528, 1226, 2936, 6822, 16212, 38081, 90717, 215414, 516358

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

Antti Karttunen, Counted with the help of this C program.

Bisections: A081151, A081152.
Cf. A081157, A081160, A081162, A057507, A081148.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081157(n) + A081160(n) + A081162(n) = A057507(n) - A081148(n).

A085197 Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.

Original entry on oeis.org

1, 3, 6, 8, 11, 15, 17, 20, 22, 25, 29, 31, 34, 38, 43, 45, 48, 50, 53, 57, 59, 62, 64, 67, 71, 73, 76, 80, 85, 87, 90, 92, 95, 99, 101, 104, 108, 113, 115, 118, 122, 127, 133, 135, 138, 140, 143, 147, 149, 152, 154, 157, 161, 163, 166, 170, 175, 177, 180, 182, 185, 189

Offset: 1

Antti Karttunen, Jun 14 2003. Proposed by Wouter Meeussen Mar 15 2003

nonn

From the second term 3 onward also one more than the partial sums of A076050.

Paolo Xausa, Table of n, a(n) for n = 1..16796
Antti Karttunen, Notes concerning A080237-tree and related sequences.

Cf. A085196. First column of A085180.

PositionIndex[Nest[Flatten[Map[Range[#+1] &, #]] &, {1}, 6]][[1]] (* Paolo Xausa, Mar 04 2024 *)

a(n) = A080336(n-1) + n = A082854(A082315(A072795(A081291(n-1)))).

a(n) = n if n < 2, otherwise a(n-1)+A076050(n-1).

A089410 Least common multiple of all cycle sizes (also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A074679/A074680.

Original entry on oeis.org

1, 1, 2, 5, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78

Offset: 0

Antti Karttunen, Nov 29 2003

nonn

A. Karttunen, C-program for computing the initial terms of this sequence

Cf. A016825.

A086587 Least common multiple of cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutations A085169/A085170.

Original entry on oeis.org

1, 1, 1, 2, 10, 90, 1260, 167580, 10345048560, 210224307704851440, 142378995493242911206243440, 4409130655192711420325660927780160, 308972448405145190275995459920449062174478109373358971999360

Offset: 0

Antti Karttunen, Jun 23 2003

A089848 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089851/A089853.

Original entry on oeis.org

1, 1, 2, 2, 5, 15, 42, 132, 431, 1430, 4862, 16801, 58786, 208012, 742914, 2674440, 9694845, 35357712, 129644790, 477638700, 1767263322, 6564120420, 24466267020, 91482564069, 343059613650, 1289904147324, 4861946402882, 18367353072152

Offset: 0

Antti Karttunen, Nov 29 2003

The number of n-node binary trees fixed by the corresponding automorphism(s).

A. Karttunen, C-program for computing the initial terms of this sequence

A089848 := n -> A000108(n-1)+`if`((2 = (n mod 3)), A000108((n-2)/3),0);

(define (A089848 n) (cond ((zero? n) 1) (else (+ (A000108 (- n 1)) (if (= 2 (modulo n 3)) (A000108 (/ (- n 2) 3)) 0)))))

a(0)=1, a(n) = A000108(n-1) if (n mod 3)=0 or 1 and a(n) = A000108(n-1)+A000108((n-2)/3) if (n mod 3)=2.

A081148 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 0, 1, 0, 4, 0, 9, 0, 24, 0, 62, 0, 162, 0, 447, 0, 1234, 0

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

Antti Karttunen, Counted with the help of this C program.

Bisection: A081149.
Cf. A081153, A081155, A057507, A081150.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081153(n) + A081155(n) = A057507(n) - A081150(n).

A081151 Number of even cycles in range [A014137(2n-1)..A014138(2n-1)] of permutation A057505/A057506.

Original entry on oeis.org

0, 1, 3, 18, 95, 528, 2936, 16212, 90717, 516358, 2979992

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Equal to the number of all cycles in the same range (after n>0).

Cf. A081152.
Cf. A081150, A081158, A001700, A057507.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081150(2n) = A081158(n)+A001700(n-1) [where A001700(-1) is taken as 0 here] = A057507(2n) [After n>0].

A081157 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with no fixed points of either A057163 or A057164.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 8, 20, 60, 148, 402, 986, 2474, 5918, 14496, 34708, 84282, 202664, 492048

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Cf. A014137, A014138, A057505, A057506, A057163, A057164

Bisections: A081158, A081159. Cf. A081150, A081160, A081162.

cp's OEIS Frontend

A074080 Triangle T(n,k) (listed in order T(1,0), T(2,0), T(2,1), T(3,0), T(3,1), T(3,2), T(4,0), etc.) giving the number of 2^k-cycles that occur in the n-th sub-permutation of A069767/A069768 (i.e., in the range [A014137(n-1)..A014138(n-1)] inclusive).

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A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

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A081150 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

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A085197 Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.

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A089410 Least common multiple of all cycle sizes (also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A074679/A074680.

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A086587 Least common multiple of cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutations A085169/A085170.

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A089848 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089851/A089853.

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A081148 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

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A081151 Number of even cycles in range [A014137(2n-1)..A014138(2n-1)] of permutation A057505/A057506.

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A081157 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with no fixed points of either A057163 or A057164.

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