A277101 -id:A277101 - cp's OEIS frontend

A276428 Sum over all partitions of n of the number of distinct parts i of multiplicity i.

0, 1, 0, 1, 2, 3, 3, 6, 7, 12, 15, 22, 27, 40, 49, 68, 87, 116, 145, 193, 239, 311, 387, 494, 611, 776, 952, 1193, 1464, 1817, 2214, 2733, 3315, 4060, 4911, 5974, 7195, 8713, 10448, 12585, 15048, 18039, 21486, 25660, 30462, 36231, 42888, 50820, 59972, 70843, 83354

Offset: 0

Emeric Deutsch, Sep 19 2016

nonn

			a(5) = 3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1',2',2], [1,1,3], [2,3], [1',4], [5] of 5 only the marked parts satisfy the requirement.

Vaclav Kotesovec, Table of n, a(n) for n = 0..7500 (terms 0..1000 from Alois P. Heinz)
Philip Cuthbertson, David J. Hemmer, Brian Hopkins, and William J. Keith, Partitions with fixed points in the sequence of first-column hook lengths, arXiv:2401.06254 [math.CO], 2024.

Cf. A276427, A276434, A277101; A114638, A116861.

g := (sum(x^(i^2)*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==j, x, 1]*b[n - i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; a[n_] := (row = T[n]; row.Range[0, Length[row]-1]); Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Nov 28 2016, after Alois P. Heinz's Maple code for A276427 *)

apply( A276428(n,s,c)={forpart(p=n,c=1;for(i=1,#p,p[i]==if(i<#p, p[i+1])&&c++&&next; c==p[i]&&s++; c=1));s}, [0..20]) \\ M. F. Hasler, Oct 27 2019

a(n) = Sum_{k>=0} k*A276427(n,k).

G.f.: g(x) = Sum_{i>=1} (x^{i^2}*(1-x^i))/Product_{i>=1} (1-x^i).

A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 5, 6, 10, 12, 19, 23, 34, 41, 58, 72, 98, 121, 162, 200, 262, 323, 415, 511, 650, 796, 1000, 1222, 1522, 1851, 2287, 2771, 3399, 4103, 5000, 6015, 7289, 8735, 10530, 12579, 15094, 17968, 21468, 25477, 30319, 35873, 42531, 50177, 59291

Offset: 0

Emeric Deutsch, Sep 30 2016

nonn

			a(6) = 3 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2], [1',1,2,2], [2',2,2], [1,1,1,3], [1,2,3], [3,3], [1',1,4], [2,4], [1,5], [6] of 6 only the marked parts satisfy the requirement.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A276427, A276428, A276429, A276433, A277099, A277100, A277101, A277102.

g := (sum(x^(i*(i+1))*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i+1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016

max = 60; s = Sum[x^(i*(i+1))*(1-x^i), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

a(n) = Sum(k*A276433(n,k), k>=0).

G.f.: g(x) = Sum_(i>=1) (x^(i(i+1))(1-x^i))/Product_(i>=1) (1-x^i).

A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 1, 6, 1, 8, 3, 12, 3, 18, 3, 1, 24, 6, 32, 10, 45, 10, 1, 59, 17, 1, 79, 21, 1, 104, 28, 3, 137, 37, 2, 177, 50, 4, 229, 64, 4, 295, 82, 8, 377, 105, 8, 477, 139, 10, 1, 605, 174, 13, 761, 220, 21, 956, 275, 24, 1193, 350, 31, 1

Offset: 0

Emeric Deutsch, Sep 30 2016

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277099(n).
Sum(k*T(n,k), k>=0) = A276434(n).

			The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
T(5,1) = 1, counting [1,1,3].
T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
T(8,2) = 1, counting [1,1,2,2,2].
Triangle starts:
1;
1;
1,1;
3;
4,1;
6,1;
8,3.

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.

G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n],t,k),k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 30 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).

A277099 Number of partitions of n containing no part i of multiplicity i+1.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 8, 12, 18, 24, 32, 45, 59, 79, 104, 137, 177, 229, 295, 377, 477, 605, 761, 956, 1193, 1484, 1840, 2276, 2800, 3441, 4210, 5141, 6261, 7603, 9206, 11132, 13419, 16144, 19380, 23223, 27763, 33134, 39467, 46931, 55703, 66008, 78085, 92239, 108776, 128091, 150617

Offset: 0

Emeric Deutsch, Sep 30 2016

nonn

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A276427, A276428, A276429, A276433, A276434, A277100, A277101, A277102.

g:= product(1/(1-x^i)-x^(i*(i+1)), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i+1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016

nmax = 50; CoefficientList[Series[Product[(1/(1-x^k) - x^(k*(k+1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 30 2016 *)

a(n) = A276433(n,0).

G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^(i*(i+1))).

A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 2, 7, 4, 10, 5, 15, 6, 1, 21, 8, 1, 28, 13, 1, 37, 18, 1, 50, 25, 2, 67, 31, 3, 88, 42, 5, 115, 55, 6, 150, 73, 8, 193, 93, 11, 248, 122, 15, 317, 154, 19, 402, 200, 24, 1, 508, 253, 30, 1, 640, 320, 41, 1, 802, 399, 53, 1, 1002, 503, 69, 1

Offset: 0

Emeric Deutsch, Oct 10 2016

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277102(n).
Sum(k*T(n,k), k>=0) = A277101(n).

			The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
T(5,1) = 2, counting [1,1,1,2] and [2,3].
T(8,2) = 1, counting [2,3,3].
Triangle starts:
1;
1;
1, 1;
2, 1;
4, 1;
5, 2;
7, 4;
...

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A116608, A276427, A276428, A276429, A276433, A276434, A277099, A277101, A277102.

g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): gser := simplify(series(g, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i-1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Oct 10 2016

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, x, 1]*b[n-i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 08 2016 after Alois P. Heinz *)

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^(i+1))).

A277102 Number of partitions of n containing no part i of multiplicity i-1.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 10, 15, 21, 28, 37, 50, 67, 88, 115, 150, 193, 248, 317, 402, 508, 640, 802, 1002, 1248, 1545, 1908, 2351, 2887, 3532, 4313, 5251, 6377, 7724, 9334, 11254, 13541, 16253, 19473, 23286, 27791, 33100, 39362, 46723, 55370, 65504, 77377, 91257, 107477, 126380

Offset: 0

Emeric Deutsch, Oct 10 2016

nonn

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277101.

g := (product(1/(1-x^(i+1))-x^(i*(i+1)), i = 1 .. 100))/(1-x): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i-1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, 0, b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 11 2016 after Alois P. Heinz *)

a(n) = A277100(n,0).

G.f.: g(x) = Product_{i>=1}(1/(1-x^(i+1)) - x^(i(i+1))).

A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Oct 29 2019

In the n-th row, the partitions of n are considered in the "Abramowitz and Stegun" or graded (reflected or not) colexicographic ordering, as in A036036 or A036037. For each partition this counts the numbers m > 0 such that there are exactly m parts equal to m in the partition.

Row lengths are A000041(n) = number of partitions of n, the partition numbers.

			The table reads:
  n \ T(n,k), ...
  0 : 0;   (The only partition of 0 is [], having no number at all in it.)
  1 : 1;   (The only partition of 1 is [1], in which the number m=1 occurs 1 time.)
  2 : 0,0;   (Neither [2] nor [1,1] have some m occurring m times.)
  3 : 0,1,0;   ([3] and [1,1,1] have no m, but [1,2] has m=1 occurring m times.)
  4 : 0,1,1,0,0;   (Here [1,3] and [2,2] have m=1 resp. m=2 occurring m times.)
  5 : 0,1,0,0,2,0,0;   ([1,4] has m=1, [1,2,2] has m=1 and m=2 occurring m times.)
  6 : 0,1,0,0,0,1,0,0,1,0,0;
  7 : 0,1,0,0,0,1,1,1,0,0,1,0,1,0,0;
  (...)
Column 1 = (0,1,0,...) = A063524, characteristic function of {1}: The corresponding partition is [n], except for [] when n=0.
Column 2 = (0,1,1,1,...) = signum(n-2) = A057427(n-2), n >= 2: The corresponding partition is [1, n-1].
Column 3 = A063524(n-3) = A185014(n), characteristic function of {4}: The corresponding partition is [2, n-2] for n >= 4, and [1,1,1] for n = 3.
Column 4 = (0,...) = A000004(n-4), the zero function: The corresponding partition is [3, n-3] for n >= 6, and [1,1,2] for n = 4 and [1,1,3] for n = 5.
Row sums = A276428(n) = sum over all partitions of n of the number of distinct parts m of multiplicity m.

OEIS Wiki, Orderings of partitions

Cf. A036036 (list of partitions in Abramowitz & Stegun or graded reflected colexicographic order).
Cf. A000041 (partition numbers = row lengths).
Cf. A063524 (col.1: chi_{1}), A057427 (col.2: signum), A185014 (col.3: chi_{4}), A000004 (col.4: zero).
Cf. A276427 (frequency of 0, ..., max.value in each row), A276428 (row sums), A276429, A276434, A277101.
Cf. A328806 (row length of A276427(n) = 1 + largest value in row n).

apply( A328891_row(n, r=[])={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); r=concat(r,s));r}, [0..12])

cp's OEIS Frontend

A276428 Sum over all partitions of n of the number of distinct parts i of multiplicity i.

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A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

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A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

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A277099 Number of partitions of n containing no part i of multiplicity i+1.

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A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

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A277102 Number of partitions of n containing no part i of multiplicity i-1.

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A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.

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