A076276 -id:A076276 - cp's OEIS frontend

A182699 Number of emergent parts in all partitions of n.

0, 0, 0, 0, 1, 1, 4, 4, 10, 12, 22, 27, 47, 56, 89, 112, 164, 205, 294, 364, 505, 630, 845, 1052, 1393, 1719, 2235, 2762, 3533, 4343, 5506, 6730, 8443, 10296, 12786, 15531, 19161, 23161, 28374, 34201, 41621, 49975, 60513, 72385, 87200, 103999, 124670, 148209

Offset: 0

Omar E. Pol, Nov 29 2010

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n.
Also, here the "filler parts" of the partitions of n are defined to be the parts of the last section of the set of partitions of n that are not the emergent parts.
For n >= 4, length of row n of A183152. - Omar E. Pol, Aug 08 2011
Also total number of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

			For n = 6 the partitions of 6 contain four "emergent" parts: (3), (4), (2), (2), so a(6) = 4. See below the location of the emergent parts.
6
(3) + 3
(4) + 2
(2) + (2) + 2
5 + 1
3 + 2 + 1
4 + 1 + 1
2 + 2 + 1 + 1
3 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
For a(10) = 22 see the link for the location of the 22 "emergent parts" (colored yellow and green) and the location of the 42 "filler parts" (colored blue) in the last section of the set of partitions of 10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740.

b:= proc(n, i) option remember; local t, h;
      if n<0 then [0, 0, 0]
    elif n=0 then [0, 1, 0]
    elif i<2 then [0, 0, 0]
    else t:= b(n, i-1); h:= b(n-i, i);
         [t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]
      fi
    end:
a:= n-> b(n, n)[3]:
seq (a(n), n=0..50);  # Alois P. Heinz, Oct 21 2011

b[n_, i_] := b[n, i] = Module[{t, h}, Which[n<0, {0, 0, 0}, n == 0, {0, 1, 0}, i<2 , {0, 0, 0}, True, t = b[n, i-1]; h = b[n-i, i]; Join [t[[1]] + h[[1]] + h[[2]], t[[2]], t[[3]] + h[[3]] + h[[1]] ]]]; a[n_] := b[n, n][[3]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)

a(n) = A138135(n) - A002865(n), n >= 1.
From Omar E. Pol, Oct 21 2011: (Start)
a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1.
a(n) = A138137(n) - A000041(n), n >= 1. (End)
a(n) = A076276(n) - A006128(n-1), n >= 1. - Omar E. Pol, Oct 30 2011

A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 5, 13, 38, 116, 382, 1310, 4748, 17848, 70076, 284252, 1195240, 5174768, 23103368, 105899656, 498656912, 2404850720, 11879332048, 59976346448, 309442319456, 1628921941312, 8746095288800, 47840221880288, 266492604100288, 1510338372987776

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of set partitions of [n] in which the block containing 1 is of length <= 3 and all other blocks are of length <= 2. Example: a(4)=13 counts all 15 partitions of [4] except 1234 and 1/234. - David Callan, Jul 22 2008

Empirical: a(n) is the sum of the entries in the second-last row of the lower-triangular matrix of coefficients giving the expansion of degree-(n+1) complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

			G.f. = x + 2*x + 5*x^2 + 13*x^3 + 38*x^4 + 116*x^5 + 382*x^6 + 1310*x^7 + ... - _Michael Somos_, Jan 23 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86 (divided by 2).
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).

John Cerkan, Table of n, a(n) for n = 1..795
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.

Cf. A000085, A001189, A013989, A076276, A248475.

a:=[1, 2];; for n in [3..10^2] do a[n] := a[n-1] + n*a[n-2]; od; a;  # Muniru A Asiru, Jan 25 2018

I:=[1,2]; [n le 2 select I[n] else Self(n-1)+n*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 31 2018

a := proc(n) option remember: if n = 1 then 1 elif n = 2 then 2 elif  n >= 3 then procname(n-1) +n*procname(n-2) fi; end:
seq(a(n), n = 1..100); # Muniru A Asiru, Jan 25 2018

RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1]+n a[n-2]},a,{n,30}] (* Harvey P. Dale, Apr 21 2012 *)
(* Programs from Michael Somos, Jan 23 2018 *)
a[n_]:= With[{m=n+1}, If[m<2, 0, Sum[(2 k-1)!! Binomial[m, 2 k], {k, 0, m/2}]/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricU[-m/2, 1/2, -1/2] / (-1/2)^(m/2)/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricPFQ[{-m/2, (1-m)/2}, {}, 2]/2]];
a[n_]:= If[ n<1, 0, n! SeriesCoefficient[Exp[x+x^2/2]*(1+x)/2, {x, 0, n}]]; (* End *)
Fold[Append[#1, #1[[-1]] + #2 #1[[-2]]] &, {1, 2}, Range[3, 26]] (* Michael De Vlieger, Jan 23 2018 *)

{a(n) = if( n<1, 0, n! * polcoeff( exp( x + x^2/2 + x * O(x^n)) * (1 + x) / 2, n))}; /* Michael Somos, Jan 23 2018 */

my(N=30,x='x+O('x^N)); Vec(serlaplace((1/2)*( (1+x)*exp(x + x^2/2) - 1))) \\ Joerg Arndt, Sep 04 2023

def A001475_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( ((1+x)*exp(x+x^2/2) -1)/2 ).egf_to_ogf().list()
a=A001475_list(40); a[1:] # G. C. Greubel, Sep 03 2023

a(n) = (1/2)*A000085(n+1).
E.g.f.: (1/2)*( (1+x)*exp(x + x^2/2) - 1). - Vladeta Jovovic, Nov 04 2003
Given e.g.f. y(x), then 0 = y'(x) * (1+x) - (y(x)+1/2) * (2+2*x+x^2) = 1 - y''(x) + y'(x)*(1 + x) + 2*y(x). - Michael Somos, Jan 23 2018
0 = +a(n)*(+a(n+1) +a(n+2) -a(n+3)) +a(n+1)*(-a(n+1) +a(n+2)) for all n>0. - Michael Somos, Jan 23 2018
a(n) ~ n^((n+1)/2) / (2^(3/2) * exp(n/2 - sqrt(n) + 1/4)) * (1 + 19/(24*sqrt(n))). - Vaclav Kotesovec, Apr 01 2018

More terms from Harvey P. Dale, Apr 21 2012

A152534 Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0} (1-x^k/faq(k,q)).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 3, 1, 5, 7, 11, 11, 8, 4, 1, 7, 13, 25, 36, 44, 42, 36, 24, 13, 5, 1, 11, 24, 54, 93, 142, 184, 215, 222, 208, 172, 126, 81, 44, 19, 6, 1, 15, 39, 98, 195, 344, 532, 753, 964, 1150, 1264, 1294, 1226, 1082, 880, 661, 451, 278, 151, 70, 26, 7, 1

Offset: 0

Vladeta Jovovic, Dec 06 2008

			Triangle begins:
  1;
  1;
  2,  1;
  3,  3,  3,  1;
  5,  7, 11, 11,  8,  4,  1;
  7, 13, 25, 36, 44, 42, 36, 24, 13,  5,  1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A005651 (row sums), A000041 (first column), A076276 (second column), A152474, A152536.

T(n,n) gives A346980.

multinomial2q := proc(n::integer,k::integer,nparts::integer)
        local lpar ,res, constrp;
        res := [] ;
        if n< 0 or nparts <= 0 then
                ;
        elif nparts = 1 then
                if n = k then
                        return [[n]] ;
                end if;
        else
                for lpar from 0 do
                        if lpar*nparts > n or lpar > k then
                                break;
                        end if;
                        for constrp in procname(n-nparts*lpar,k-lpar,nparts-1) do
                                if nops(constrp) > 0 then
                                        res := [op(res),[op(constrp),lpar]] ;
                                end if;
                        end do:
                end do:
        end if ;
        return res ;
end proc:
multinomial2 := proc(n::integer,k::integer)
        local res,constrp ;
        res := [] ;
        for constrp in multinomial2q(n,k,n) do
                if nops(constrp) > 0 then
                        res := [op(res),constrp] ;
                end if ;
        end do:
        res ;
end proc:
faq := proc(i,q)
        mul((q^j-1)/(q-1),j=1..i) ;
end proc;
A152534 := proc(n,k)
        pi := [] ;
        for sp from 0 to n do
                pi := [op(pi),op(multinomial2(n,sp))] ;
        end do;
        tqk := 0 ;
        for p in pi do
                faqe :=1 ;
                for i from 1 to nops(p) do
                        faqe := faqe* faq(i,q)^op(i,p) ;
                end do:
                tqk := tqk+faq(n,q)/faqe ;
        end do;
        tqk ;
        coeftayl(tqk,q=0,k) ;
end proc:
for n from 1 to 8 do
        for k from 0 to binomial(n,2) do
                printf("%d,",A152534(n,k)) ;
        end do;
        printf("\n") ;
end do: # R. J. Mathar, Sep 27 2011
# second Maple program:
f:= proc(n) option remember; `if`(n<2, 1, f(n-1)*(q^n-1)/(q-1)) end:
b:= proc(n, i) option remember; simplify(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)/f(i)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(simplify(f(n)*b(n$2))):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 09 2021

f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
b[n_, i_] := b[n, i] = Simplify[If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]/f[i]^j, {j, 0, n/i}]]];
T[n_] := CoefficientList[Simplify[f[n]*b[n, n]], q];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} faq(n,q)/Product_{i=1..n} faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.
Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = 1.
Sum_{k=0..binomial(n,2)} (-1)^k*T(n,k) = A152536(n). - Alois P. Heinz, Aug 09 2021

T(0,0)=1 prepended by Alois P. Heinz, Aug 09 2021

A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 4, 6, 9, 12, 17, 22, 30, 39, 51, 65, 85, 107, 136, 171, 216, 268, 335, 413, 512, 629, 772, 941, 1151, 1396, 1694, 2046, 2471, 2969, 3569, 4271, 5110, 6093, 7258, 8620, 10235, 12113, 14325, 16902, 19925, 23434, 27540, 32296, 37842, 44260, 51715, 60322, 70306, 81805

Offset: 1

Wouter Meeussen, Oct 07 2014

nonn

Empirical: a(n) is the number of zeros in the subdiagonal of the lower-triangular matrix of coefficients giving the expansion of degree-n complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

			The successor pair (3,1,1,1) and (2,2,2) are incomparable in dominance ordering, and so are their transposes (4,1,1) and (3,3) and these are the two only pairs for n=6, hence a(6)=2.

Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.

Wikipedia, Dominance Order

Cf. A182988, A248476, A001475, A076276.

Needs["Combinatorica`"];
dominant[par1_?PartitionQ,par2_?PartitionQ]:= Block[{le=Max[Length[par1],Length[par2]],acc},
acc=Accumulate[PadRight[par1,le]]-Accumulate[PadRight[par2,le]];Which[Min[acc]===0&&Max[acc]>=0,1,Min[acc]<=0&&Max[acc]===0,-1,True,0]];
Table[Count[Apply[dominant, Partition[Partitions[n], 2,1], 1],0], {n,40}]

A161981 A006128(n) mod n.

Original entry on oeis.org

0, 1, 0, 0, 0, 5, 5, 6, 2, 2, 0, 3, 10, 10, 3, 7, 10, 16, 2, 10, 4, 1, 17, 22, 1, 20, 24, 21, 11, 23, 28, 31, 30, 4, 16, 14, 18, 3, 4, 26, 29, 9, 42, 8, 15, 5, 29, 43, 38, 18, 32, 32, 22, 1, 3, 3, 28, 21, 32, 51, 30, 46, 39, 19, 52, 16, 30, 1, 2, 68, 65, 70, 57, 73, 42, 1, 21, 25, 44, 72, 17, 71

Offset: 1

Juri-Stepan Gerasimov, Jun 23 2009

nonn

The remainder of (the total number of parts in all partitions of n) mod n.

			a(1)=0=1 mod 1. a(2)=1=3 mod 2. a(3)=0=6 mod 3. a(4)=0=12 mod 4. a(5)=0=20 mod 5. a(6)=5=35 mod 6.

Cf. A000041, A006128, A076276.

Edited, a 3 inserted, and extended by R. J. Mathar, Aug 02 2009

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A182699 Number of emergent parts in all partitions of n.

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A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

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A152534 Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0} (1-x^k/faq(k,q)).

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A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.

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A161981 A006128(n) mod n.

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