Christian Stump - cp's OEIS frontend

A298248 Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.

1, 1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 10, 2, 10, 0, 1, 1, 0, 20, 12, 54, 12, 20, 0, 1, 1, 0, 35, 42, 212, 140, 212, 42, 35, 0, 1, 1, 0, 56, 112, 675, 880, 1592, 880, 675, 112, 56, 0, 1, 1, 0, 84, 252, 1845, 3962, 9246, 9540, 9246, 3962, 1845, 252, 84, 0, 1

Offset: 0

Christian Stump, Jan 16 2018

DE(n,k) = number of permutations with d descents and e descents of the inverse such that d+e = k.

			The triangle DE(n, k) begins:
n\k 0    1     2     3      4      5      6     7     8    9   10
0:  1
1:  1
2:  1    0     1
3:  1    0     4     0      1
4:  1    0    10     2     10      0      1
5:  1    0    20    12     54     12     20     0     1
6:  1    0    35    42    212    140    212    42    35    0    1

Christian Stump, On bijections between 231-avoiding permutations and Dyck paths, MathSciNet:2734176

Dominique Foata and Guo-Niu Han, The q-series in Combinatorics; permutation statistics
FindStat - Combinatorial Statistic Finder, The sum of the number of descents and the number of recoils of a permutation

Row sums give A000142.

Cf. A000292, A008292, A180389.

q = var("q")
[sum( q^(pi.number_of_descents()+pi.inverse().number_of_descents()) for pi in Permutations(n) ).coefficients(sparse=False) for n in [1 .. 6]]

A273254 Dimensions of odd-dimensional spheres with unique smooth structure.

Original entry on oeis.org

1, 3, 5, 61

Offset: 1

Christian Stump, May 18 2016

Guozhen Wang, Zhouli Xu, On the uniqueness of the smooth structure of the 61-sphere, arXiv:1601.02184 [math.AT], 2016.

A264049 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of integer partitions lambda of n such that there are k partitions mu such that the Gelfand-Tsetlin polytope for lambda and mu is integral.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 3, 0, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 3, 2, 2, 2, 0, 2, 0, 3, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1

Offset: 1

Christian Stump, Nov 02 2015

Row sums give A000041, n >= 1.

			Triangle begins:
1,
1,1,
1,1,1,
1,1,1,1,1,
1,2,0,2,0,1,1,
1,2,1,2,0,1,1,1,0,1,1,
1,3,0,2,2,1,1,1,0,0,1,1,0,1,1,
1,3,2,2,2,0,2,0,3,0,0,2,0,0,1,0,1,0,1,0,1,1,
...

FindStat - Combinatorial Statistic Finder, Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
J. De Loera and T. B. McAllister, Vertices of Gelfand-Tsetlin polytopes, arXiv:math/0309329 [math.CO], 2003, MathSciNet:2096742.

Cf. A000041, A264035, A264047, A264048.

A264048 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of integer partitions lambda of n such that there are k compositions mu such that the Gelfand-Tsetlin polytope for lambda and mu is integral.

Original entry on oeis.org

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

Offset: 1

Christian Stump, Nov 02 2015

Row sums give A000041, n >= 1.

			Triangle begins:
1,
1,1,
1,0,1,1,
1,0,0,1,1,0,1,1,
1,0,0,0,1,0,1,0,0,0,1,1,0,0,1,1,
1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,0,0,2,0,0,0,0,1,1,
...

FindStat - Combinatorial Statistic Finder, Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
J. De Loera and T. B. McAllister, Vertices of Gelfand-Tsetlin polytopes, arXiv:math/0309329 [math.CO], 2003, MathSciNet:2096742.

Cf. A000041, A264035, A264047, A264049.

A264047 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of integer partitions lambda of n such that there are k compositions mu such that the Gelfand-Tsetlin polytope for lambda and mu is non-integral.

Original entry on oeis.org

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

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

			Triangle begins:
1,
1,
2,
3,
5,
5,2,
6,3,0,1,0,0,1,
7,1,0,0,0,0,0,2,2,0,0,1,0,0,0,0,1,1,
8,1,0,0,0,0,0,0,2,0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,2,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,
...

FindStat - Combinatorial Statistic Finder, Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
J. De Loera and T. B. McAllister, Vertices of Gelfand-Tsetlin polytopes, arXiv:math/0309329 [math.CO], 2003, MathSciNet:2096742.

Cf. A000041, A264035, A264048, A264049.

A264035 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of integer partitions lambda of n such that there are k partitions mu such that the Gelfand-Tsetlin polytope for lambda and mu is non-integral.

Original entry on oeis.org

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

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

			Triangle begins:
1,
1,
2,
3,
5,
5,2,
6,4,1,
7,2,3,2,1,
8,2,3,2,3,3,0,1,
...

FindStat - Combinatorial Statistic Finder, Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
J. De Loera and T. B. McAllister, Vertices of Gelfand-Tsetlin polytopes, arXiv:math/0309329 [math.CO], 2003, MathSciNet:2096742.

Cf. A000041, A264047, A264048, A264049.

A264032 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of integer partitions of n containing exactly k partitions.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 1, 2, 2, 0, 0, 4, 1, 2, 0, 0, 2, 2, 2, 2, 1, 2, 0, 0, 0, 0, 2, 2, 2, 4, 1, 0, 2, 2, 0, 0, 0, 0, 0, 4, 0, 0, 4, 3, 0, 2, 2, 2, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 2, 4, 0, 0, 4, 1, 0, 4, 2, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 4, 2, 0, 0, 2, 2, 4, 0, 0, 6, 2, 0, 4

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

			Triangle begins:
1,
1,
2,
2,1,
2,1,2,
2,0,0,4,1,
2,0,0,2,2,2,2,1,
2,0,0,0,0,2,2,2,4,1,0,2,
2,0,0,0,0,0,4,0,0,4,3,0,2,2,2,2,0,1,
2,0,0,0,0,0,0,2,0,0,3,2,4,0,0,4,1,0,4,2,2,0,2,2,
...

FindStat - Combinatorial Statistic Finder, The number of partitions contained in the given partition.

Cf. A000041.

A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4).
Also the number of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2). - Alois P. Heinz, Nov 14 2015

			Triangle begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,    1;
4 :  14,    8,    2;
5 :  42,   45,   25,    7,   1;
6 : 132,  220,  198,  112,  44,  12,   2;
7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1;
...

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
FindStat - Combinatorial Statistic Finder, The number of nestings of a permutation, The number of crossings of a permutation
R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.

Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497.

Cf. A000142, A001754, A002620, A260665, A260670, A287328, A291722.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 14 2015

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^(j-1)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[ T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

T(n,n) = A287328(n). - Alois P. Heinz, Aug 31 2017

More terms from Alois P. Heinz, Oct 26 2015

A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.
The weighted sum is given by the sum of the rows where row i is weighted by i.
Note that the first part has weight 0. This statistic (zero-based weighted sum) is ranked by A359677, reverse A359674. Also the number of partitions of n with one-based weighted sum n + k. - Gus Wiseman, Jan 10 2023

			Triangle T(n,k) begins:
  1;
  1;
  1,1;
  1,1,0,1;
  1,1,1,1,0,0,1;
  1,1,1,1,1,0,1,0,0,0,1;
  1,1,1,2,1,0,2,1,0,0,1,0,0,0,0,1;
  1,1,1,2,1,1,2,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1;
  1,1,1,2,2,1,2,2,1,1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1;
  ...
The a(15,31) = 5 partitions of 15 with weighted sum 31 are: (6,2,2,1,1,1,1,1), (5,4,1,1,1,1,1,1), (5,2,2,2,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1). These are also the partitions of 15 with one-based weighted sum 46. - _Gus Wiseman_, Jan 09 2023

Alois P. Heinz, Rows n = 0..50, flattened
FindStat - Combinatorial Statistic Finder, Weighted size of a partition

Row sums are A000041.
The version for compositions is A053632, ranked by A124757 (reverse A231204).
Row lengths are A152947, or A161680 plus 1.
The one-based version is also A264034, if we use k = n..n(n+1)/2.
The reverse version A358194 counts partitions by sum of partial sums.
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
A359678 counts multisets by zero-based weighted sum.
Cf. A029931, A066185, A124757, A304818, A318283, A337206, A359042.

b:= proc(n, i, w) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, w)+
      `if`(i>n, 0, x^(w*i)*b(n-i, i, w+1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 01 2015

b[n_, i_, w_] := b[n, i, w] = Expand[If[n == 0, 1, If[i < 1, 0, b[n, i - 1, w] + If[i > n, 0, x^(w*i)*b[n - i, i, w + 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==k&]],{n,0,8},{k,n,n*(n+1)/2}] (* Gus Wiseman, Jan 09 2023 *)

From Alois P. Heinz, Jan 20 2023: (Start)
max_{k=0..A161680(n)} T(n,k) = A337206(n).
Sum_{k=0..A161680(n)} k * T(n,k) = A066185(n). (End)

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 0, 2, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 7, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 7, 3, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

Column k=0 gives A025065(n-2) for n>=2.

			Triangle begins:
0,1,
0,1,
1,1,
1,2,
2,2,1,
2,3,0,2,
4,2,1,1,1,1,1,
4,3,1,0,0,2,2,0,1,0,1,0,0,0,1,
7,2,0,0,1,0,3,0,1,0,2,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
...

Alois P. Heinz, Rows n = 0..14, flattened
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
FindStat - Combinatorial Statistic Finder, Number of standard Young tableaux of an integer partition such that no k and k+1 appear in the same row.

Cf. A000041, A001181, A237770, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, x^h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+ `if`(i>n, 0,
       g(n-i, i, [i, l[]]))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 02 2015

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := If[n == 0 || i == 1, x^h[Join[Array[1 &, n], l], 0], If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Join[{i}, l]]] ]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Sum_{k=1..A264078(n)} k*T(n,k) = A237770(n). - Alois P. Heinz, Nov 02 2015

cp's OEIS Frontend

User: Christian Stump

A298248 Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.

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A273254 Dimensions of odd-dimensional spheres with unique smooth structure.

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A264049 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of integer partitions lambda of n such that there are k partitions mu such that the Gelfand-Tsetlin polytope for lambda and mu is integral.

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A264048 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of integer partitions lambda of n such that there are k compositions mu such that the Gelfand-Tsetlin polytope for lambda and mu is integral.

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A264047 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of integer partitions lambda of n such that there are k compositions mu such that the Gelfand-Tsetlin polytope for lambda and mu is non-integral.

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A264035 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of integer partitions lambda of n such that there are k partitions mu such that the Gelfand-Tsetlin polytope for lambda and mu is non-integral.

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A264032 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of integer partitions of n containing exactly k partitions.

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A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

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A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

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