A000701 -id:A000701 - cp's OEIS frontend

A067136 Number of Young tableaux with n cells whose shape is symmetric.

1, 1, 0, 2, 2, 6, 16, 20, 132, 112, 1216, 1440, 12440, 25520, 138048, 476320, 1649312, 9138300, 21842944, 182232248, 345145392, 3805004296, 7002149760, 83299368432, 180168275232, 1907968553776, 5402826994176, 45597877829600

Offset: 0

Naohiro Nomoto, Feb 19 2002

nonn

Equivalently, the row lengths are a self-conjugate partition of n.

			For n = 8; 4+2+1+1 produces 90 tableaux and 3+3+2 produces
42 tableaux. so a(8) = 90+42 = 132.
For n = 5 = 3+1+1 the 6 tableaux are:
123..124..125..135..134..145
4....3....3....2....2....2..
5....5....4....4....5....3..

Sean A. Irvine, Table of n, a(n) for n = 0..100
Sean A. Irvine, Java program (github)

Cf. A000085, A000700, A000701, A067142, A330645.

a(n) = A000085(n) - 2*A067142(n).

a(n) = A000085(n) - A330645(n). - Omar E. Pol, Jan 11 2020

Edited by Franklin T. Adams-Watters, Nov 07 2006

A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 8, 10, 10, 12, 14, 15, 18, 20, 22, 26, 29, 32, 36, 40, 44, 50, 56, 60, 68, 76, 82, 92, 101, 110, 122, 134, 146, 160, 176, 191, 210, 230, 248, 272, 296, 320, 350, 380, 410, 446, 484, 522, 566, 612, 660, 715, 772, 830, 896, 966, 1038, 1120

Offset: 0

Naohiro Nomoto, Feb 24 2002

Also number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k (Dean Hickerson). Absolute values of the terms of A053254. - Emeric Deutsch, Feb 10 2006
The number of self-conjugate partitions of n into odd parts is nonzero if and only if n = 4*k + 1 for some nonnegative integer k. - Michael Somos, Jul 25 2015
Also number of C3v plane partitions of n = 3*k + 1 with rank 1 ; equivalently number of self-conjugate integer partitions with (weight-length) = n. - Wouter Meeussen, May 23 2025

			a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
G.f. = 1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + ...

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. S. Andersen, A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n), arXiv:2007.09794 [math.CO], 2020.
George E. Andrews, The Bhargava-Adiga Summation and Partitions, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.

Cf. A000700, A000701.

Cf. A053253, A053254, A085140, A047993.

g:=sum(q^(k*(k+1))/product(1-q^(2*j+1),j=0..k),k=0..8): gser:=series(g,q=0,80): seq(coeff(gser,q,n),n=0..75); # Emeric Deutsch, Feb 10 2006

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / Product[ 1 - x^i, {i, 1, 2 k + 1, 2}], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jul 25 2015 *)
Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* Wouter Meeussen, May 24 2025 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))}; /* Michael Somos, Jan 27 2008 */

/* Continued Fraction Expansion: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + (-x)^(n-k+1)*(1 - (-x)^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013

G.f.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - Emeric Deutsch and Dean Hickerson
G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - Dean Hickerson and Vladeta Jovovic
G.f.: 1/(1 - x*(1+x)/(1 + x^2*(1-x^2)/(1 - x^3*(1+x^3)/(1 + x^4*(1-x^4)/(1 - x^5*(1+x^5)/(1 - ...)))))), a continued fraction. - Paul D. Hanna, Jul 09 2013
From Michael Somos, Jul 25 2015: (Start)
Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.
a(n) = (-1)^n * A053254(n).
a(2*n) = A085140(n).
a(2*n + 1) = A053253(n). (End)
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

More terms from Emeric Deutsch, Feb 10 2006

A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 1, 1, 12, 2, 1, 16, 5, 1, 20, 9, 1, 25, 16, 1, 30, 25, 1, 36, 39, 1, 1, 42, 56, 2, 1, 49, 80, 5, 1, 56, 109, 10, 1, 64, 147, 19, 1, 72, 192, 32, 1, 81, 249, 54, 1, 90, 315, 84, 1, 100, 396, 129, 1, 1, 110, 489, 190, 2, 1, 121, 600, 275, 5

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1   1
   1   2
   1   4
   1   6
   1   9   1
   1  12   2
   1  16   5
   1  20   9
   1  25  16
   1  30  25
   1  36  39   1
   1  42  56   2
   1  49  80   5
   1  56 109  10
For example, row n = 7 counts the following partitions:
  (1111111)  (7)       (43)
             (52)      (331)
             (61)
             (322)
             (421)
             (511)
             (2221)
             (3211)
             (4111)
             (22111)
             (31111)
             (211111)

Row sums are A000041.
Row lengths are A000194, reversed A003056.
Column k = 1 is A002620, reversed A238875.
Column k = 2 is A097701.
The version for permutations is A008292, opposite A123125.
The weak version is A115720/A115994, rank statistic A257990.
The version for compositions is A352524, weak A352525.
The version for reversed partitions is A353319.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

partsabove[y_]:=Length[Select[Range[Length[y]],#

A067142 One half of the number of Young tableaux with n cells whose shape is asymmetric.

Original entry on oeis.org

0, 0, 1, 1, 4, 10, 30, 106, 316, 1254, 4140, 17128, 63856, 271492, 1126216, 4936608, 22278712, 101330506, 487735440, 2313734596, 11706759352, 58073844300, 305941244576, 1587272257096, 8656011151184, 46886237603400, 263791190603200, 1487539434072976

Offset: 0

Naohiro Nomoto, Feb 19 2002

nonn

Equivalently, the row lengths are a non-self-conjugate partition of n.

			a(4) = 4 = 8/2; the 8 tableaux are:
1..1234..123..124..134..14..12..13
2........4....3....2....2...3...2.
3.......................3...4...4.
4.................................
The two tableaux of size 4 with symmetric shape are excluded:
12..13
34..24

Cf. A000085, A000700, A000701, A067136, A330645.

a(n) = (A000085(n) - A067136(n))/2.

a(n) = A330645(n)/2. - Omar E. Pol, Jan 11 2020

Edited by Franklin T. Adams-Watters, Nov 07 2006

a(26)-a(27) from Omar E. Pol, Jan 11 2020

A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 6, 1, 3, 7, 2, 5, 9, 2, 1, 8, 11, 2, 1, 13, 14, 1, 3, 19, 15, 3, 5, 27, 19, 1, 11, 34, 22, 2, 1, 15, 49, 23, 2, 1, 27, 59, 28, 3, 3, 39, 78, 30, 1, 5, 60, 93, 34, 3, 11, 82, 118, 36, 1, 18, 115, 140, 41, 3, 1, 30, 155, 170, 42, 2, 1, 48

Offset: 1

Naohiro Nomoto, Feb 02 2002

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
..........1.1.3.5..8.13.19.27.34.49....etc
...................1..1..3..5.11.15....etc

Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.

compareL := proc(L1,L2)
    if nops(L1) < nops(L2) then
        -1 ;
    elif nops(L1) > nops(L2) then
        1;
    else
        for i from 1 to nops(L1) do
            if op(i,L1) > op(i,L2) then
                return 1 ;
            elif op(i,L1) < op(i,L2) then
                return -1 ;
            end if;
        end do:
        0 ;
    end if;
end proc:
A067627 := proc(n,k)
    local a,p,s,pc ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(s) = k then
            pc := combinat[conjpart](p) ;
            if compareL(p,pc) <= 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:
for n from 1 to 30 do
for k from A003056(n) to 1 by -1 do
    printf("%4d,",A067627(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 08 2019

More terms from R. J. Mathar, May 08 2019

A067772 One half of the number of non-self-conjugate balanced partitions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 12, 17, 19, 26, 31, 40, 47, 61, 72, 91, 108, 134, 159, 197, 231, 283, 335, 405, 477, 576, 676, 810, 950, 1131, 1325, 1572, 1834, 2166, 2527, 2970, 3455, 4051, 4702, 5493, 6366, 7412, 8574, 9959, 11493, 13315, 15345, 17729, 20392

Offset: 1

Naohiro Nomoto, Feb 06 2002

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000700, A000701, A047993, A331262.

seq(n) = {my(A=O(x*x^n), m=1); while(3*m^2-m <= 2*n, m++); Vec(1 + sum(k=1, m-1, (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ), A )/eta(x + A) - prod(k=1, (n+1)\2, 1 + x^(2*k-1) + A), -n)/2} \\ Andrew Howroyd, Apr 20 2021

a(n) = (A047993(n) - A000700(n))/2.

a(n) = A331262(n)/2. - Omar E. Pol, Jun 18 2022

Terms a(37) and beyond from Andrew Howroyd, Apr 20 2021

A096648 Number of partitions of an n-set with odd number of even blocks.

Original entry on oeis.org

0, 1, 3, 7, 25, 106, 434, 2045, 10707, 57781, 338195, 2115664, 13796952, 95394573, 692462671, 5235101739, 41436754261, 341177640610, 2915100624274, 25866987547865, 237448494222575, 2252995117706961, 22078799199129799, 222971522853648704, 2319210969809731600

Offset: 1

Vladeta Jovovic, Aug 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A000701.

Cf. A024429, A024430, A096647.

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, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, Mod[t+If[Mod[i, 2] == 0, j, 0], 2]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
With[{nn=30},Rest[CoefficientList[Series[Exp[Sinh[x]]Sinh[Cosh[x]-1], {x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 03 2016 *)

E.g.f.: exp(sinh(x))*sinh(cosh(x)-1).
a(2*n) = A024429(2*n) and a(2*n+1) = A024430(2*n+1). - Jonathan Vos Post, Oct 19 2005
a(n) = sum{k=0..n, if(mod(n-k,2)=1, A048993(n,k), 0)}. - Paul Barry, May 19 2006

More terms from Emeric Deutsch, Nov 16 2004

A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 2, 1, 5, 4, 2, 7, 6, 2, 10, 6, 6, 15, 7, 7, 1, 18, 14, 7, 3, 26, 15, 11, 4, 35, 17, 19, 6, 47, 24, 19, 11, 61, 33, 22, 18, 1, 80, 44, 28, 20, 4, 103, 54, 42, 25, 7, 138, 60, 57, 31, 11, 175, 85, 58, 52, 15, 224, 112, 66, 64, 24

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1  1
   2  1
   2  3
   4  2  1
   5  4  2
   7  6  2
  10  6  6
  15  7  7  1
  18 14  7  3
  26 15 11  4
  35 17 19  6
  47 24 19 11
  61 33 22 18  1
  80 44 28 20  4
For example, row n = 9 counts the following reversed partitions:
  (1134)       (9)     (27)   (234)
  (1224)       (18)    (36)
  (1233)       (117)   (45)
  (11115)      (126)   (135)
  (11124)      (1116)  (144)
  (11133)      (1125)  (225)
  (11223)      (2223)  (333)
  (12222)
  (111114)
  (111123)
  (111222)
  (1111113)
  (1111122)
  (11111112)
  (111111111)

Row sums are A000041.
Row lengths are A003056.
The version for permutations is A008292, opposite A123125.
The weak unreversed version is A115720/A115994, rank statistic A257990.
For fixed points instead of excedances we have A238352, rank stat A352822.
Column k = 0 is A238875.
The version for compositions is A352524, weak A352525.
The version for unreversed partitions is A353318.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
Cf. A000701, A002620, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

partsabove[y_]:=Length[Select[Range[Length[y]],#

A064480 Form a conjugate partition of row with 1+1+1 in first row. all other rows are the union of their parents. a(n) = number of types of piles in the n-th row.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 19, 26, 36, 51, 69, 94, 130, 188, 261, 366, 514, 710, 993, 1399, 1995, 2779, 3912, 5490, 7723, 10848, 15230, 21457, 30165, 42401, 59718, 83808, 117844, 165932, 233358, 328316, 461885, 650105, 915243, 1287795, 1812815, 2552260, 3593697

Offset: 1

Naohiro Nomoto, Feb 14 2002

nonn

The n-th row sum is equal to 3*2^(n-1).

The largest part of the n-th row is A000204(n).

			Start with 1+1+1 from which a(1)=1.
The conjugate of 1+1+1 is 3, giving the union 3+1+1+1, and a(2)=2.
The conjugate of 3+1+1+1 is 4+1+1, giving the union 4+3+1+1+1+1+1, and a(3)=3.
The conjugate of 4+3+1+1+1+1+1 is 7+2+2+1, giving the union 7+4+3+2+2+1+1+1+1+1+1, and a(4)=5.

Sean A. Irvine, Java program (github)

Cf. A000700, A000701, A000204.

More terms from Sean A. Irvine, Jul 13 2023

A064660 The number of distinct parts in the partition sequence lambda(n) formed by the recurrence lambda(1) = 1 and lambda(n+1) is the sum of lambda(n) and its conjugate.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 15, 22, 30, 39, 53, 75, 106, 151, 215, 297, 424, 592, 835, 1162, 1618, 2274, 3217, 4556, 6361, 8940, 12560, 17645, 24822, 34812, 48967, 68861, 96939, 136462, 191896, 269976, 379726, 534239, 751829, 1058170, 1489038, 2096243, 2951262

Offset: 1

Naohiro Nomoto, Feb 14 2002

nonn

lambda(n) is a partition of 2^(n-1).
The largest part of lambda(n) is A000045(n).
The number of parts of lambda(n) is A000045(n+1). Peter J. Taylor, Jul 24 2014

			lambda(4) = 3+2+1+1+1 has conjugate partition 5+2+1, so lambda(5) = 5+3+2+2+1+1+1+1 and a(5) = |{5,3,2,1}| = 4.

Cf. A000700, A000701, A000045.

More terms, description and example rephrased by Peter J. Taylor, Jul 24 2014

cp's OEIS Frontend

A067136 Number of Young tableaux with n cells whose shape is symmetric.

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A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

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A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

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Mathematica

A067142 One half of the number of Young tableaux with n cells whose shape is asymmetric.

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A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

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Maple

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A067772 One half of the number of non-self-conjugate balanced partitions.

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A096648 Number of partitions of an n-set with odd number of even blocks.

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Maple

Mathematica

Formula

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A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

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