A056971 -id:A056971 - cp's OEIS frontend

A323963 Number of defective (binary) heaps on n elements with exactly seven defects.

0, 210, 7616, 185460, 4200960, 77887920, 1310267904, 21025461600, 314958758400, 4765553385120, 69086614338560, 1013373907056960, 14714554645140480, 212868974122848000, 3168605413206097920, 45787509175434048000, 705241846739554713600, 10297540417675899801600

Offset: 7

Alois P. Heinz, Feb 09 2019

nonn

Or number of permutations p of [n] having exactly seven indices i in {1,...,n} such that p(i) > p(floor(i/2)).

Alois P. Heinz, Table of n, a(n) for n = 7..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Column k=7 of A306343.

Cf. A056971.

A323964 Number of defective (binary) heaps on n elements with exactly eight defects.

Original entry on oeis.org

0, 896, 32460, 1201920, 29987760, 705522048, 14197154400, 264121228800, 4765553385120, 80958196300800, 1386261729792960, 22797317814635520, 378396940680172800, 6210934579834490880, 101592763596409536000, 1694705349980956262400, 27653450672540750745600

Offset: 8

Alois P. Heinz, Feb 09 2019

nonn

Or number of permutations p of [n] having exactly eight indices i in {1,...,n} such that p(i) > p(floor(i/2)).

Alois P. Heinz, Table of n, a(n) for n = 8..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Column k=8 of A306343.

Cf. A056971.

A323965 Number of defective (binary) heaps on n elements with exactly nine defects.

Original entry on oeis.org

0, 3360, 190800, 6837600, 239978112, 6268548000, 153775564800, 3365570435040, 69086614338560, 1386261729792960, 26344715667079680, 502986050203680000, 9251596509192069120, 171657357483560256000, 3142069546022259916800, 57414248711716723200000

Offset: 9

Alois P. Heinz, Feb 09 2019

nonn

Or number of permutations p of [n] having exactly nine indices i in {1,...,n} such that p(i) > p(floor(i/2)).

Alois P. Heinz, Table of n, a(n) for n = 9..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Column k=9 of A306343.

Cf. A056971.

A323966 Number of defective (binary) heaps on n elements with exactly ten defects.

Original entry on oeis.org

0, 19200, 947760, 49680576, 1744142400, 60169824000, 1646888944800, 42633973724160, 1013373907056960, 22797317814635520, 502986050203680000, 10556482426015426560, 222685725334400064000, 4535040919258899456000, 92995172484979478630400, 1880492299058130490163200

Offset: 10

Alois P. Heinz, Feb 09 2019

nonn

Or number of permutations p of [n] having exactly ten indices i in {1,...,n} such that p(i) > p(floor(i/2)).

Alois P. Heinz, Table of n, a(n) for n = 10..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Column k=10 of A306343.

Cf. A056971.

A324062 Number of defective (binary) heaps on n elements where one ancestor-successor pair does not have the correct order.

Original entry on oeis.org

0, 0, 1, 2, 6, 16, 60, 240, 840, 3584, 16800, 96000, 475200, 3041280, 19219200, 153753600, 864864000, 6560153600, 47048601600, 439934976000, 3192583680000, 31434670080000, 280947363840000, 3296449069056000, 27139515346944000, 308787374614118400

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly one pair (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

			a(4) = 6: 3241, 3412, 3421, 4123, 4132, 4213.
a(5) = 16: 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241.
(The examples use max-heaps.)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=1 of A306393.

Cf. A056971.

b:= proc(u, o) option remember; local n, g, l; n:= u+o;
      if n=0 then 1
    else g:= 2^ilog2(n); l:= min(g-1, n-g/2); expand(
         add(x^(n-j)*add(binomial(j-1, i)*binomial(n-j, l-i)*
         b(i, l-i)*b(j-1-i, n-l-j+i), i=0..min(j-1, l)), j=1..u)+
         add(x^(j-1)*add(binomial(j-1, i)*binomial(n-j, l-i)*
         b(l-i, i)*b(n-l-j+i, j-1-i), i=0..min(j-1, l)), j=1..o))
      fi
    end:
a:= n-> coeff(b(n, 0), x, 1):
seq(a(n), n=0..25);

b[u_, o_] := b[u, o] = Module[{n, g, l}, n = u + o; If[n == 0, 1,
     g = 2^(Length[IntegerDigits[n, 2]]-1); l = Min[g-1, n-g/2]; Expand[
     Sum[ x^(n - j)*Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*
     b[i, l-i]*b[j-1-i, n-l-j+i], {i, 0, Min[j - 1, l]}], {j, 1, u}] +
     Sum[x^(j-1)*Sum[Binomial[j-1, i]*Binomial[n - j, l - i]*
     b[l-i, i]*b[n-l-j+i, j-1-i], {i, 0, Min[j-1, l]}], {j, 1, o}]]]];
a[n_] := Coefficient[b[n, 0], x, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

A324063 Number of defective (binary) heaps on n elements where two ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 2, 6, 24, 100, 480, 1890, 8960, 47040, 288000, 1584000, 10644480, 74131200, 615014400, 3783780000, 29520691200, 230015385600, 2199674880000, 17239951872000, 172890685440000, 1660143513600000, 19778694414336000, 174145223476224000, 2007117934991769600

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly two pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=2 of A306393.

Cf. A056971.

A324064 Number of defective (binary) heaps on n elements where three ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 6, 24, 120, 640, 3150, 16128, 94080, 614400, 3801600, 26864640, 203174400, 1757184000, 11783772000, 95122227200, 794598604800, 7821066240000, 65767223808000, 675845406720000, 6895980748800000, 83909612666880000, 784784318782464000

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly three pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=3 of A306393.

Cf. A056971.

A324065 Number of defective (binary) heaps on n elements where four ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 3, 24, 120, 720, 4200, 24192, 151200, 1056000, 7286400, 54743040, 442041600, 3997593600, 29081052000, 244365721600, 2164235673600, 21996748800000, 197620929792000, 2090405560320000, 22475789107200000, 280198170869760000, 2772753817946112000

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly four pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=4 of A306393.

Cf. A056971.

A324066 Number of defective (binary) heaps on n elements where five ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 0, 16, 120, 720, 4830, 31360, 211680, 1555200, 11880000, 95293440, 815443200, 7687680000, 60432372000, 530552422400, 4945330790400, 51912327168000, 496766020608000, 5425624055808000, 61093281300480000, 781258429366272000, 8157685988035584000

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly five pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=5 of A306393.

Cf. A056971.

A324067 Number of defective (binary) heaps on n elements where six ancestor-successor pairs do not have the correct order.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 100, 720, 5040, 36736, 268800, 2073600, 17186400, 147502080, 1331616000, 13047091200, 110053944000, 1011903692800, 9874978713600, 106953080832000, 1086116967936000, 12275238666240000, 144074916311040000, 1890064025321472000

Offset: 0

Alois P. Heinz, Feb 13 2019

nonn

Or number of permutations p of [n] having exactly six pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
Wikipedia, Permutation

Column k=6 of A306393.

Cf. A056971.

cp's OEIS Frontend

A323963 Number of defective (binary) heaps on n elements with exactly seven defects.

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A323964 Number of defective (binary) heaps on n elements with exactly eight defects.

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A323965 Number of defective (binary) heaps on n elements with exactly nine defects.

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A323966 Number of defective (binary) heaps on n elements with exactly ten defects.

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A324062 Number of defective (binary) heaps on n elements where one ancestor-successor pair does not have the correct order.

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A324063 Number of defective (binary) heaps on n elements where two ancestor-successor pairs do not have the correct order.

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A324064 Number of defective (binary) heaps on n elements where three ancestor-successor pairs do not have the correct order.

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A324065 Number of defective (binary) heaps on n elements where four ancestor-successor pairs do not have the correct order.

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A324066 Number of defective (binary) heaps on n elements where five ancestor-successor pairs do not have the correct order.

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A324067 Number of defective (binary) heaps on n elements where six ancestor-successor pairs do not have the correct order.

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