A000085 -id:A000085 - cp's OEIS frontend

A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 6, 1, 0, 1, 1, 2, 4, 9, 10, 1, 0, 1, 1, 2, 4, 10, 21, 20, 1, 0, 1, 1, 2, 4, 10, 25, 51, 35, 1, 0, 1, 1, 2, 4, 10, 26, 70, 127, 70, 1, 0, 1, 1, 2, 4, 10, 26, 75, 196, 323, 126, 1, 0, 1, 1, 2, 4, 10, 26, 76, 225, 588, 835, 252, 1, 0

Offset: 0

Alois P. Heinz, Apr 16 2012

Also the number A(n,k) of standard Young tableaux of n cells and <= k columns.

A(n,k) is also the number of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,ak), where #(z,x) counts the letters x in word z. The A(4,4) = 10 words of length 4 over alphabet {a,b,c,d} are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca, abcd.

			A(4,2) = 6, there are 6 standard Young tableaux of 4 cells and height <= 2:
  +------+  +------+  +---------+  +---------+  +---------+  +------------+
  | 1  3 |  | 1  2 |  | 1  3  4 |  | 1  2  4 |  | 1  2  3 |  | 1  2  3  4 |
  | 2  4 |  | 3  4 |  | 2 .-----+  | 3 .-----+  | 4 .-----+  +------------+
  +------+  +------+  +---+        +---+        +---+
Square array A(n,k) begins:
  1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  2,   2,   2,   2,   2,   2,   2, ...
  0,  1,  3,   4,   4,   4,   4,   4,   4, ...
  0,  1,  6,   9,  10,  10,  10,  10,  10, ...
  0,  1, 10,  21,  25,  26,  26,  26,  26, ...
  0,  1, 20,  51,  70,  75,  76,  76,  76, ...
  0,  1, 35, 127, 196, 225, 231, 232, 232, ...
  0,  1, 70, 323, 588, 715, 756, 763, 764, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened
Wikipedia, Young tableau

Columns k=0-12 give: A000007, A000012, A001405, A001006, A005817, A049401, A007579, A007578, A007580, A212915, A212916, A229053, A229068.
Main diagonal gives A000085.
A(2n,n) gives A293128.
Cf. A047884, A049400, A226873, A240608.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..15);

h[l_List] := Module[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_List] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
a[n_, k_] := g[n, k, {}];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *)

Conjecture: A(n,k) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * Product_{j=1..k} Gamma(j/2). - Vaclav Kotesovec, Sep 12 2013

A111924 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,1) for n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 0, 1, 6, 3, 0, 1, 10, 15, 0, 0, 1, 15, 45, 15, 0, 0, 1, 21, 105, 105, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 1, 66, 1485, 13860, 51975, 62370

Offset: 1

N. J. A. Sloane, Nov 25 2005

T(n,k) = number of partitions of an n-set into k nonempty subsets, each of size at most 2.

			Triangle begins:
1
1 1
1 3 0
1 6 3 0
1 10 15 0 0
1 15 45 15 0 0
1 21 105 105 0 0 0
1 28 210 420 105 0 0 0
1 36 378 1260 945 0 0 0 0

J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.

A100861 is another version of this triangle. Row sums give A000085.

T[n_, 0] = 0; T[1, 1] = 1; T[2, 1] = 1; T[n_, k_] := T[n - 1, k - 1] + (n - 1)T[n - 2, k - 1]; Table[T[n, k], {n, 12}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)

The Choi-Smith reference gives many further properties and formulas.

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).

More terms from Robert G. Wilson v, Dec 09 2005

A001189 Number of degree-n permutations of order exactly 2.

Original entry on oeis.org

0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503, 2390479, 10349535, 46206735, 211799311, 997313823, 4809701439, 23758664095, 119952692895, 618884638911, 3257843882623, 17492190577599, 95680443760575, 532985208200575, 3020676745975551

Offset: 1

N. J. A. Sloane

Number of set partitions of [n] into blocks of size 2 and 1 with at least one block of size 2. - Olivier Gérard, Oct 29 2007

For n>=2, number of standard Young tableaux with height <= n - 1. That is, all tableaux (A000085) but the one with just one column. - Joerg Arndt, Oct 24 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..800
N. Chheda, M. K. Gupta, RNA as a Permutation, arXiv:1403.5477 [q-bio.BM], 2014.
R. B. Herrera, The number of elements of given period in finite symmetric group, Amer. Math. Monthly 64, 1957, 488-490.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
J. Rangel-Mondragon, Selected Themes in Computational Non-Euclidean Geometry: Part 1, The Mathematica Journal 15 (2013); http://www.mathematica-journal.com/data/uploads/2013/07/Rangel-Mondragon_Selected-1.pdf
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., April 2004.

Cf. A001470-A001473, A052501, A053496-A053504, A061121-A061128.
Column k=1 of A143911, column k=2 of A080510, A182222. - Alois P. Heinz, Oct 24 2012
Column k=2 of A057731. - Alois P. Heinz, Feb 14 2013

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2) -Exp(x) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, May 14 2019

a:= proc(n) option remember; `if`(n<3, [0$2, 1][n+1],
      a(n-1) +(n-1) *(1+a(n-2)))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 24 2012
# alternative:
A001189 := proc(n)
    local a,prs,p,k ;
    a := 0 ;
    for prs from 1 to n/2 do
        p := product(binomial(n-2*k,2),k=0..prs-1) ;
        a := a+p/prs!;
    end do:
    a;
end proc:
seq(A001189(n),n=1..13) ; # R. J. Mathar, Jan 04 2017

RecurrenceTable[{a[1]==0,a[2]==1,a[n]==a[n-1]+(1+a[n-2])(n-1)},a[n],{n,25}] (* Harvey P. Dale, Jul 27 2011 *)

{a(n) = sum(j=1,floor(n/2), n!/(j!*(n-2*j)!*2^j))}; \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x +x^2/2) - exp(x), x, 0, m); a=[factorial(n)*T.coefficient(x, n) for n in (0..m)]; a[1:] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^2/2) - exp(x).
a(n) = A000085(n) - 1.
a(n) = b(n, 2), where b(n, d)=Sum_{k=1..n} (n-1)!/(n-k)! * Sum_{l:lcm{k, l}=d} b(n-k, l), b(0, 1)=1 is the number of degree-n permutations of order exactly d.
From Henry Bottomley, May 03 2001: (Start)
a(n) = a(n-1) + (1 + a(n-2))*(n-1).
a(n) = Sum_{j=1..floor(n/2)} n!/(j!*(n-2*j)!*(2^j)). (End)

A014068 a(n) = binomial(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

Offset: 0

N. J. A. Sloane

nonn

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..370
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

[Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)

from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024

[(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

A321765 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 0, -1, 1, -1, 1, 2, 1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 1, 1, 0, 1, -1, -1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, 0, -1, 1, 0, 0, -1

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
   1  -1
   1   1
   1  -1   1
   1   0  -1
   1   0  -1   1  -1
   1   2   1
   1   2  -1  -1   1
   1  -1   0   0   1
   1  -1   0   0   1  -1   1
   1   0   1  -1  -1
   1   0  -1   0   0   1   0   0  -1   1  -1
   1   0  -1   1   0   0  -1
For example, row 12 gives: p(211) = s(4) + s(31) - s(211) - s(1111).

Wikipedia, Symmetric polynomial

Row sums are A317554.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.

A001472 Number of degree-n permutations of order dividing 4.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 6224, 33616, 218656, 1326656, 9893632, 70186624, 574017536, 4454046976, 40073925376, 347165733632, 3370414011904, 31426411211776, 328454079574016, 3331595921852416, 37125035407900672, 400800185285464064, 4744829049220673536

Offset: 0

N. J. A. Sloane and J. H. Conway

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..570 (first 201 terms from T. D. Noe)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 25 (Dead link)
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A053495.

Column k=4 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^4/4) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

spec := [S, {S = Set(Union(Cycle(Z, card = 1), Cycle(Z, card = 2), Cycle(Z, card = 4)))}, labeled]; seq(combstruct[count](spec, size = n), n = 0 .. 23); # David Radcliffe, Aug 29 2025

n = 23; CoefficientList[Series[Exp[x+x^2/2+x^4/4], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011 *)

a(n):=n!*sum(sum(binomial(k,j)*binomial(j,n-4*k+3*j)*(1/2)^(n-4*k+3*j)*(1/4)^(k-j),j,floor((4*k-n)/3),k)/k!,k,1,n); /* Vladimir Kruchinin, Sep 07 2010 */

my(N=33, x='x+O('x^N)); egf=exp(x+x^2/2+x^4/4); Vec(serlaplace(egf)) /* Joerg Arndt, Sep 15 2012 */

m = 30; T = taylor(exp(x + x^2/2 + x^4/4), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^2/2 + x^4/4).
D-finite with recurrence: a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(n) = a(n-1) + (n-1)*a(n-2) + (n^3-6*n^2+11*n-6)*a(n-4) for n>3. - H. Palsdottir (hronn07(AT)ru.is), Sep 19 2008
a(n) = n!*Sum_{k=1..n} (1/k!)*Sum_{j=floor((4*k-n)/3)..k} binomial(k,j) * binomial(j,n-4*k+3*j) * (1/2)^(n-4*k+3*j)*(1/4)^(k-j), n>0. - Vladimir Kruchinin, Sep 07 2010
a(n) ~ n^(3*n/4)*exp(n^(1/4)-3*n/4+sqrt(n)/2-1/8)/2 * (1 - 1/(4*n^(1/4)) + 17/(96*sqrt(n)) + 47/(128*n^(3/4))). - Vaclav Kotesovec, Aug 09 2013

A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 4, 4, 5, 5, 6, 1, 1, 5, 5, 5, 5, 9, 9, 10, 10, 16, 1, 1, 6, 6, 14, 14, 14, 14, 15, 15, 20, 21, 21, 35, 35, 1, 1, 7, 7, 14, 14, 20, 20, 21, 21, 28, 28, 35, 35, 42, 56, 56, 64, 64, 70, 70, 90, 1, 1, 8, 8, 27, 27, 28, 28, 42, 42, 42, 48, 48, 56, 56, 70, 84

Offset: 0

N. J. A. Sloane, Mar 21 2001

Sum_{k>=1} T(n,k)^2 = n!. - R. J. Mathar, May 09 2013
From Emeric Deutsch, Oct 31 2014: (Start)
Number of entries in row n = A000041(n) = number of partitions of n.
Sum of entries in row n = A000085(n).
Largest (= last) entry in row n = A003040(n).
The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End)

			Triangle begins:
  1;
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3, 3;
  1, 1, 4, 4, 5, 5, 6;
  ...

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New York.

Alois P. Heinz, Rows n = 0..36, flattened
J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group Canad. J. Math, 6:316-324, 1954. See Theorem 1, p. 318.
Index entries for sequences related to groups

Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247.
Maximal entry in each row gives A003040.
Cf. A000041, A000085, A000142, A060437, A093784, A224653.

CharacterTable(SymmetricGroup(6)); // (say)

h:= proc(l) local n; n:= nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
                 seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= n-> sort([g(n, n, [])])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 07 2013

h[l_List] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, Flatten @ Table[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
T[n_] := Sort[g[n, n, {}]]; T[1] = {1};
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

More terms from Vladeta Jovovic, May 20 2003

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 11, 12, 16, 12, 32, 28, 31, 8, 64, 31, 128, 33, 82, 64, 256, 28, 69, 144, 69, 86, 512, 105, 1024, 16, 208, 320, 209, 82, 2048, 704, 512, 86, 4096, 318, 8192, 216, 262, 1536, 16384, 64, 465, 262, 1232, 528, 32768, 209, 588, 245, 2912, 3328

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 11 tableaux:
1 1
1 1
.
2 1   1 1   1 1   1 2
1 1   1 2   2 2   1 2
.
1 1   1 2   1 2   1 3
2 3   1 3   3 3   2 3
.
1 2   1 3
3 4   2 4

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]Table[PrimePi[p],{k}]]]];
Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]

A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 17, 28, 41, 75, 122, 192, 314, 484, 771, 1216, 1861, 2848, 4395, 6610, 10037

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Jan 17 2019: (Start)
Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
  6   51   42   321
.
  5   4   41   31   32   31   22   221   211
  1   2   1    2    1    11   2    1     11
.
  3   21   21   111
  2   2    11   11
  1   1    1    1
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{2},{1,2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{1,1},{1,1,1}}
   {{1,2},{1,2,2}}
   {{2,2},{1,2,2}}

Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A000085, A138178, A323436.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* Gus Wiseman, Jan 18 2019 *)

a(11)-a(17) from Gus Wiseman, Jan 18 2019

a(18)-a(21) from Robert Price, Jun 21 2021

A053505 Number of degree-n permutations of order dividing 30.

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 540, 3060, 20700, 145980, 1459800, 13854600, 140059800, 1514748600, 15869034000, 285268878000, 4109761962000, 59488383690000, 935767530036000, 13364309726748000, 240338216104020000, 4540941256642020000, 79739974380153240000

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=30 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +x^2/2 +x^3/3 +x^5/5 +x^6/6 +x^10/10 +x^15/15 +x^30/30) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 5, 6, 10, 15, 30])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 5, 6, 10, 15, 30}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^5/5 +x^6/6 + x^10/10 +x^15/15 +x^30/30], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^2/2 +x^3/3 +x^5/5 + x^6/6 +x^10/10 +x^15/15 +x^30/30) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^5/5 +x^6/6 +x^10/10 +x^15/15 +x^30/30), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^2/2 + x^3/3 + x^5/5 + x^6/6 + x^10/10 + x^15/15 + x^30/30).

cp's OEIS Frontend

A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A111924 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,1) for n >= 1.

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A001189 Number of degree-n permutations of order exactly 2.

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A014068 a(n) = binomial(n*(n+1)/2, n).

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A321765 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and s is Schur functions.

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A001472 Number of degree-n permutations of order dividing 4.

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A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

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