A046682 -id:A046682 - cp's OEIS frontend

A096647 Number of partitions of an n-set with even number of even blocks.

1, 1, 1, 2, 8, 27, 97, 443, 2095, 10440, 58194, 340375, 2097933, 13847485, 95504749, 690495874, 5245040408, 41428115543, 340899165549, 2917641580783, 25857170687507, 237421321934176, 2253720620740362, 22073206655954547, 222987346441156585, 2319379362420267753

Offset: 0

Vladeta Jovovic, Aug 14 2004

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A046682, A096648.

with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1,
      0, add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
      irem(t+`if`(irem(i, 2)=0, j, 0), 2)), j=0..n/i)))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

a[n_] := Sum[If[Mod[n-k, 2] == 0, StirlingS2[n, k], 0], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 16 2015, after Paul Barry *)

E.g.f.: exp(sinh(x))*cosh(cosh(x)-1).

a(n) = sum{k=0..n, if(mod(n-k,2)=0, A048993(n,k), 0)}. - Paul Barry, May 19 2006

More terms from Emeric Deutsch, Nov 16 2004

A180652 Number of different hook length multisets of partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 68, 90, 118, 149, 195, 247, 316, 400, 503, 629, 792, 982, 1219, 1508, 1864, 2286, 2806, 3427, 4176, 5079, 6159, 7445, 8998, 10824, 13009, 15598, 18673, 22288, 26594, 31632, 37586, 44571, 52783, 62356, 73650, 86764, 102106, 119975, 140802

Offset: 0

Paul-Olivier Dehaye (pdehaye(AT)math.ethz.ch), Sep 14 2010

nonn

Cf. A000009, A046682, A180649, A301512.

a(n) = A046682(n) - A180649(n).

a(41)-a(52) from Alois P. Heinz, Mar 22 2018

A242101 Number of conjugacy classes of the symmetric group S_n when conjugating only by even permutations.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 44, 58, 80, 104, 138, 180, 236, 302, 390, 496, 634, 800, 1010, 1264, 1586, 1970, 2448, 3024, 3734, 4582, 5622, 6862, 8372, 10168, 12336, 14912, 18010, 21672, 26052, 31226, 37384, 44632, 53226, 63318, 75238, 89202, 105630, 124832

Offset: 1

Attila Egri-Nagy, Aug 14 2014

nonn

Cf. A242099 (by dihedral group), A000041 (by symmetric group itself), A061417 (by cyclic group).

Cf. A046682.

List([1..11], n->Size(OrbitsDomain(AlternatingGroup(IsPermGroup, n), SymmetricGroup(IsPermGroup, n), \^)));

For n >=2, a(n) = A000041(n) + A000700(n) = 2*A046682(n) [by a formula in A046682]. - Eric M. Schmidt, Aug 23 2014

More terms from Eric M. Schmidt, Aug 23 2014

A322014 Heinz numbers of integer partitions with an even number of even parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 23, 25, 31, 32, 34, 36, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 55, 57, 59, 62, 64, 67, 68, 72, 73, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 91, 92, 94, 97, 98, 99, 100, 103, 105, 109, 110, 111, 114, 115, 118

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000701, A046682, A056239, A058695, A058696, A108949, A296150, A321753.

a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while add(`if`(numtheory[pi](i[1])::odd,
      0, i[2]), i=ifactors(k)[2])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],EvenQ[Count[primeMS[#],_?EvenQ]]&]

A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

Offset: 1

Gus Wiseman, May 25 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.

Twice the row media of A321649 or A321650.
For mean instead of twice median we have A326839/A326840.
This is the conjugate version of A360005.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation in terms of Heinz numbers.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A352491 gives n minus Heinz number of conjugate.
A363220 counts partitions with same median as conjugate.
Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]

A363220 Number of integer partitions of n whose conjugate has the same median.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377

Offset: 1

Gus Wiseman, May 29 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  .  (21)  (22)  (311)  (321)   (511)    (332)     (333)
                             (411)   (4111)   (422)     (711)
                             (3111)  (31111)  (611)     (4221)
                                              (3311)    (4311)
                                              (4211)    (6111)
                                              (5111)    (51111)
                                              (41111)   (411111)
                                              (311111)  (3111111)

For mean instead of median we have A047993.
For product instead of median we have A325039, ranks A325040.
For union instead of conjugate we have A360245, complement A360244.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A000700 counts self-conjugate partitions, ranks A088902.
A046682 and A352487-A352490 pertain to excedance set.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
A352491 gives n minus Heinz number of conjugate.
Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]

A067618 Number of self-conjugate partitions of n into prime parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 1, 3, 0, 0, 0, 5, 0, 0, 1, 6, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 1, 5, 0, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 5, 0

Offset: 0

Naohiro Nomoto, Feb 01 2002

Cf. A000700, A000701, A046682.

f[0, m_, k_] := 1; f[n_, 0, k_] := If[n==0, 1, 0]; f[n_, m_, k_] := If[n<0||m<0, 0, Module[{r}, f[n, m, k]=f[n, m-1, k]+If[PrimeQ[m+k], Sum[If[PrimeQ[r+k], f[n-r(2m-r), m-r-1, k+r], 0], {r, 1, m}], 0]]]; a[n_] := f[n, Floor[n/4]+1, 0]; (* f[n, m, k] = number of self-conjugate partitions of n with parts <= m such that every part+k is prime *)

Edited by Dean Hickerson, Feb 11 2002

A181964 Sum of the sizes of normalizers of all the cyclic subgroups of Alternating Group of order n.

Original entry on oeis.org

1, 1, 6, 36, 240, 2160, 20160, 241920, 2903040, 39916800, 578793600, 9580032000, 161902540800, 3007651046400, 58845346560000, 1234444603392000, 26854400821248000, 624231436308480000, 15083992450695168000, 385614968295997440000

Offset: 1

Olivier Gérard, Apr 04 2012

For each cyclic subgroup of the Alternate group on n symbols, add the size of its normalizer (permutations leaving the subgroup invariant by conjugation).

a(7) is remarquable because it is equal to the size of Alt(8).

			Decomposing by number of cyclic subgroups * size of normalizer of subgroups
a(5) = 1*60 + 4*15 + 6*10 + 0*60 + 10*6 = 240.
a(6) = 1*360 + 8*45 + (18*20+18*20) + 8*45 + 10*36 = 2160.

Cf. A053529, A181950.

a(n) = n!/2 * A046682(n).

cp's OEIS Frontend

A096647 Number of partitions of an n-set with even number of even blocks.

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A180652 Number of different hook length multisets of partitions of n.

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A242101 Number of conjugacy classes of the symmetric group S_n when conjugating only by even permutations.

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GAP

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A322014 Heinz numbers of integer partitions with an even number of even parts.

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Maple

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A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

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A363220 Number of integer partitions of n whose conjugate has the same median.

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Mathematica

A067618 Number of self-conjugate partitions of n into prime parts.

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Mathematica

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A181964 Sum of the sizes of normalizers of all the cyclic subgroups of Alternating Group of order n.

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