A246867 -id:A246867 - cp's OEIS frontend

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

1, 2, 3, 4, 6, 5, 8, 7, 10, 12, 9, 14, 15, 24, 11, 18, 20, 21, 30, 13, 22, 27, 28, 40, 42, 16, 26, 33, 35, 36, 54, 56, 60, 17, 32, 39, 44, 45, 66, 70, 72, 84, 120, 19, 34, 48, 52, 55, 63, 78, 88, 90, 105, 108, 168, 23, 38, 51, 64, 65, 77, 96, 104, 110, 126

Offset: 1

Gus Wiseman, Feb 18 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

This sequence is a permutation of the positive integers.

			Triangle of strict partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (51) (42) (321)
        (7) (61) (43) (52) (421)
     (8) (71) (62) (53) (431) (521)
(9) (81) (72) (54) (63) (621) (531) (432).

Cf. A050376, A056239, A064547, A106400, A122111, A213925, A215366, A246867, A279065, A279614, A299755, A299757, A299758.

Row lengths give A000009.

nn=25;
FDprimeList=Select[Range[nn],MatchQ[FactorInteger[#],{{?PrimeQ,?(MatchQ[FactorInteger[2#],{{2,_}}]&)}}]&];
Table[Sort[Times@@FDprimeList[[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,Length[FDprimeList]}]

A300354 Number of enriched p-trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 8, 8, 13, 17, 54, 56, 98, 125, 195, 500, 606, 921, 1317, 1912, 2635, 6667, 7704, 12142, 16958, 24891, 33388, 47792, 106494, 126475, 195475, 268736, 393179, 523775, 750251, 979518, 2090669, 2457315, 3759380, 5066524, 7420874, 9726501, 13935546

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			The a(6) = 8 enriched p-trees with distinct leaves: 6, (42), (51), ((31)2), ((32)1), (3(21)), ((21)3), (321).

Cf. A000009, A000041, A063834, A196545, A246867, A273873, A281145, A289501, A290261, A294018, A296150, A299201, A299202, A299203, A300352, A300353, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
ept[q_]:=ept[q]=If[Length[q]===1,1,Total[Times@@@Map[ept,Join@@Function[sptn,Join@@@Tuples[Permutations/@GatherBy[sptn,Total]]]/@Select[sps[q],Length[#]>1&],{2}]]];
Table[Total[ept/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,30}]

a(n) = Sum_{i=1..A000009(n)} A299203(A246867(n,i)).

A325505 Heinz number of the set of Heinz numbers of all strict integer partitions of n.

Original entry on oeis.org

2, 3, 5, 143, 493, 62651, 26718511, 22017033127, 44220524211551, 52289759420183033963, 546407750301194131199484983, 8362548333129019658779663581495109, 1828111016191440393570169991636207115709029581, 1059934964500839879758659437301868941873808925011368355891

Offset: 0

Gus Wiseman, May 07 2019

nonn

The Heinz number of a set or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Also Heinz numbers of rows of A246867 (squarefree numbers arranged by sum of prime indices A056239).

			The strict integer partitions of 5 are {(5), (4,1), (3,2)}, with Heinz numbers {11,14,15}, with Heinz number prime(11)*prime(14)*prime(15) = 62651, so a(6) = 62651.
The sequence of terms together with their prime indices begins:
                            2: {1}
                            3: {2}
                            5: {3}
                          143: {5,6}
                          493: {7,10}
                        62651: {11,14,15}
                     26718511: {13,21,22,30}
                  22017033127: {17,26,33,35,42}
               44220524211551: {19,34,39,55,66,70}
         52289759420183033963: {23,38,51,65,77,78,105,110}
  546407750301194131199484983: {29,46,57,85,91,102,130,154,165,210}

Index entries for sequences related to Heinz numbers

Cf. A001222, A003963, A015723, A056239, A066189, A112798, A145519, A147655, A215366, A246867, A325500 (non-strict version), A325504, A325506, A325512, A325513.

Table[Times@@Prime/@(Times@@Prime/@#&/@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,7}]

a(n) = Product_{i = 1..A000009(n)} prime(A246867(n,i)).
A001221(a(n)) = A001222(a(n)) = A000009(n).
A056239(a(n)) = A147655(n).
A003963(a(n)) = A325506(n).

A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

0, 1, 2, 7, 13, 31, 66, 138, 279, 581, 1173, 2375, 4783, 9630, 19316, 38802, 77689, 155673, 311639, 623845, 1248179, 2497719, 4996387, 9995304, 19992908, 39990902, 79986136, 159983241, 319975073, 639971495, 1279962115, 2559966847, 5119970499, 10240030209

Offset: 0

Gus Wiseman, May 23 2024

nonn

			The strict partitions of 6 are (6), (5,1), (4,2), (3,2,1), with respective binary ranks 32, 17, 10, 7 with sum 66, so a(6) = 66.

Alois P. Heinz, Table of n, a(n) for n = 0..3321

Row sums of A118462 (binary ranks of strict partitions).
For Heinz number the non-strict version is A145519, row sums of A215366.
For Heinz number (not binary rank) we have A147655, row sums of A246867.
The non-strict version is A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A277905 groups all positive integers by binary rank of prime indices.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite A371572, sum A230877
- opposite complement A371571, sum A359359
Cf. A000041, A005940, A015716, A018819, A019565, A118457, A225620, A344086.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 [0, p[1]*2^(i-1)]
          +p)(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Table[Total[Total[2^(#-1)]& /@ Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

a(n) = Sum_{k=1..n} 2^(k-1) * A015716(n,k). - Alois P. Heinz, May 24 2024

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 5, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 7, 6, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 8, 7, 6, 6, 4, 4, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

			T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - _Gus Wiseman_, May 07 2019

Mircea Merca, Table of n, a(n) for n = 1..7260

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

g:=product(1+x^j,j=1..50)*sum(t^i*x^i/(1+x^i),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014

z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
a(n) = A238450(n) + A238451(n).
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 3, 2, 4, 1, 6, 4, 2, 5, 1, 3, 2, 1, 7, 4, 3, 5, 2, 6, 1, 4, 2, 1, 8, 5, 3, 6, 2, 7, 1, 4, 3, 1, 5, 2, 1, 9, 5, 4, 6, 3, 7, 2, 8, 1, 4, 3, 2, 5, 3, 1, 6, 2, 1, 10, 6, 4, 7, 3, 8, 2, 9, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 7, 2, 1, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(21)
  4: (4)(31)
  5: (5)(32)(41)
  6: (6)(42)(51)(321)
  7: (7)(43)(52)(61)(421)
  8: (8)(53)(62)(71)(431)(521)
  9: (9)(54)(63)(72)(81)(432)(531)(621)

Wikiversity, Lexicographic and colexicographic order

Starting with reversed partitions gives A026793.
The version for compositions is A124734.
Showing partitions as Heinz numbers gives A246867.
The non-strict version is A334301 (reversed: A036036).
Ignoring length gives A344086 (reversed: A246688).
Same as A344089 with partitions reversed.
The version for revlex instead of lex is A344092.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A185974, A193073, A211992, A296774, A334302, A334433, A334435, A334438, A334439, A334440, A334441, A334442, A344091.

Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 22, 29, 37, 50, 63, 81, 106, 129, 160, 203, 246, 303, 373, 449, 541, 654, 782, 932, 1116, 1322, 1559, 1848, 2167, 2537, 2978, 3470, 4041, 4706, 5449, 6303, 7291, 8402, 9665, 11117, 12744, 14592, 16708, 19062, 21730, 24757, 28141

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also omega of the product of products of parts over all strict integer partitions of n.

The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Cf. A001222, A003963, A015716, A015723, A022629, A056239, A066189, A112798, A147655, A246867, A325504, A325506.

Table[Sum[Total[PrimeOmega/@s],{s,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

a(n) = A001222(A325504(n)).

A300351 Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 11, 20, 32, 22, 25, 40, 64, 17, 44, 50, 80, 128, 34, 55, 88, 100, 160, 256, 23, 68, 110, 125, 176, 200, 320, 512, 46, 85, 121, 136, 220, 250, 352, 400, 640, 1024, 31, 92, 170, 242, 272, 275, 440, 500, 704, 800, 1280, 2048, 62, 115, 184

Offset: 1

Gus Wiseman, Mar 03 2018

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

			Triangle of partitions into odd parts begins:
                   0
                  (1)
                  (11)
                (3) (111)
               (31) (1111)
            (5) (311) (11111)
        (51) (33) (3111) (111111)
    (7) (511) (331) (31111) (1111111)
(71) (53) (5111) (3311) (311111) (11111111)

Cf. A000009, A031368, A056239, A066208, A078408, A215366, A246867, A299759, A299757, A300063, A300272.

Table[Sort[Times@@Prime/@#&/@Select[IntegerPartitions[n],And@@OddQ/@#&]],{n,0,12}]

cp's OEIS Frontend

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

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A300354 Number of enriched p-trees of weight n with distinct leaves.

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A325505 Heinz number of the set of Heinz numbers of all strict integer partitions of n.

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A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

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A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

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A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

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A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

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