John Tyler Rascoe - cp's OEIS frontend

A386891 Irregular triangle read by rows: T(n,k) is the number of compositions of n such that the maximal cardinality of C is k, where C is a subset of the set of parts such that all elements in C appear in weakly increasing order within the composition.

1, 0, 1, 0, 2, 0, 3, 1, 0, 6, 2, 0, 11, 5, 0, 21, 10, 1, 0, 39, 23, 2, 0, 74, 49, 5, 0, 139, 107, 10, 0, 271, 216, 24, 1, 0, 524, 447, 51, 2, 0, 1031, 895, 117, 5, 0, 2023, 1813, 250, 10, 0, 3998, 3630, 544, 20, 0, 7878, 7344, 1115, 46, 1, 0, 15601, 14738, 2330, 97, 2

Offset: 0

John Tyler Rascoe, Aug 06 2025

Here the set of parts of a composition is the set of all parts appearing in the composition.

			Triangle begins:
    k=0    1    2   3  4
 n=0  1,
 n=1  0,   1,
 n=2  0,   2,
 n=3  0,   3,   1,
 n=4  0,   6,   2,
 n=5  0,  11,   5,
 n=6  0,  21,  10,  1,
 n=7  0,  39,  23,  2,
 n=8  0,  74,  49,  5,
 n=9  0, 139, 107, 10,
 n=10 0, 271, 216, 24, 1,
...
The composition of n = 3 (2,1) with set of parts {1,2} has maximal subsets {1} and {2} both with all parts appearing in weakly increasing order, so (2,1) is counted under T(3,1) = 3.
The composition of n = 15 (3,1,1,2,3,5) with set of parts {1,2,3,5} has the maximal subset {1,2,5}, so (3,1,1,2,3,5) is counted under T(15,3) = 1115.

John Tyler Rascoe, Python code.

Cf. A002024 (row lengths), A011782 (row sums).

Cf. A218796, A333213, A374629, A385604.

Python
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# see links
```

A385604 Number of compositions of n such that the odd parts are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 48, 86, 162, 292, 541, 978, 1794, 3247, 5919, 10712, 19451, 35184, 63729, 115199, 208327, 376333, 679842, 1227403, 2215695, 3998408, 7214274, 13014001, 23472678, 42331028, 76330880, 137627168, 248122171, 447301570, 806312371, 1453405651

Offset: 0

John Tyler Rascoe, Aug 02 2025

			a(5) = 14 counts all compositions of n = 5 except (1,3,1) and (3,1,1) since the odd parts are not weakly increasing.
The composition of n = 13 (2,1,1,4,2,3) has odd parts (1,1,3), so it is counted under a(13) = 1794.

Cf. A000009, A011782, A032021, A098123, A289249.

A_x(N) = {my(x='x+O('x^(N+1))); Vec((1-x^2)/(1-2*x^2)/prod(i=0,N, 1-x^(2*i+1)*(1-x^2)/(1-2*x^2)))}

G.f.: (1 - x^2)/( (1 - 2*x^2) * Product_{i>=0} (1 - x^(2*i + 1) * (1 - x^2)/(1 - 2*x^2)) ).

A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

Original entry on oeis.org

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025

With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

A386497 Number of sets of lists of [n] such that one list is the largest.

Original entry on oeis.org

1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.

b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
      `if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 23 2025

With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}

E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).

A386375 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs more frequently than any other letter.

Original entry on oeis.org

1, 1, 1, 4, 17, 96, 652, 5356, 51361, 568840, 7157036, 101048454, 1582644956, 27224336244, 509883010652, 10319902635984, 224283040843745, 5205554049801528, 128430045368430484, 3354764715348964222, 92460461868234201532, 2680680433302859375630, 81542551486359310209666

Offset: 0

John Tyler Rascoe, Jul 19 2025

			a(5) = 96 counts the following words (number of permutations shown in brackets): (1,1,1,1,1) [1], (1,1,1,1,2) [5], (1,1,1,2,2) [10], (1,1,1,2,3) [20], (1,1,2,3,4) [60].

Alois P. Heinz, Table of n, a(n) for n = 0..425

Cf. A000262, A000670, A006153, A308876, A386374.

b:= proc(n, t) option remember; `if`(n=0, 1,
      add(b(n-j, t)/j!, j=1..min(n, t)))
    end:
a:= n-> n!*add(b(n-j, j-1)/j!, j=0..n):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 19 2025

B_x(N) = {my(x='x+O('x^N)); Vec(serlaplace( sum(i=0,N, x^i/(i!*(1-sum(j=1,i-1, x^j/j!))))))}

E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i-1} x^j/j!)).

A386374 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs at least as many times as any other letter.

Original entry on oeis.org

1, 1, 3, 10, 47, 276, 2022, 17606, 179391, 2093860, 27581888, 404680398, 6541528886, 115437202986, 2206844818622, 45408726154590, 1000134868827263, 23468606700087972, 584340284516996400, 15383829737201853518, 426915367401366308112, 12454073547413511363878

Offset: 0

John Tyler Rascoe, Jul 19 2025

			a(3) = 10 counts: (1,1,1), (1,1,2), (1,2,1), (1,2,3), (1,3,2), (2,1,1), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..425

Cf. A000262, A000670, A006153, A308876.

b:= proc(n, t) option remember; `if`(n=0, 1,
      add(b(n-j, t)/j!, j=1..min(n, t)))
    end:
a:= n-> n!*add(b(n-j, j)/j!, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 19 2025

A_x(N) = {my(x='x+O('x^N)); Vec(serlaplace(sum(i=0,N, x^i/(i! *(1-sum(j=1,i, x^j/j!))))))}

E.g.f.: Sum_{i>=0} x^i/(i! * (1 - Sum_{j=1..i} x^j/j!)).

A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

Original entry on oeis.org

1, 1, 4, 15, 64, 325, 1776, 11179, 72640, 489969, 3435580, 26495491, 221599104, 1893705697, 16145571820, 138299146665, 1241234863936, 12033569772769, 124055067568788, 1303750295285563, 13577876900409280, 139418829477000801, 1441311794301705964, 15537427948684769425

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.

Alois P. Heinz, Table of n, a(n) for n = 0..650

Cf. A002999, A005183, A007837, A032299, A347006, A386254.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 17 2025

terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/((i-1)!)))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / (j-1)!).

A386254 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.

Original entry on oeis.org

1, 1, 2, 6, 18, 60, 240, 1085, 5012, 23730, 121440, 685707, 4144668, 25614589, 159141892, 1012740885, 6805631232, 48872707006, 369227821608, 2853779791619, 22131042288980, 172055270717463, 1362017827326860, 11208504802237327, 96939147303239304, 875473007351905045

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).

Alois P. Heinz, Table of n, a(n) for n = 0..674

Cf. A002999, A005183, A007837, A032299, A347006, A386255.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/j!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 17 2025

terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/i!))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / j!).

A385373 Number of solid partitions with multiplicities (1, ..., n).

Original entry on oeis.org

1, 1, 6, 138, 14049, 6851919

Offset: 0

John Tyler Rascoe, Jun 27 2025

A solid partition with d distinct parts (p_1^(k_1) > p_2^(k_2) > ... > p_d^(k_d)) has the multiset of multiplicities (k_1, k_2, ..., k_d).

Alternatively, a(n) is the number of chains of plane partitions ordered by inclusion, comprised of n consecutive triangular numbers starting with 1.

			For n = 2 a solid partition having multiplicities (1,2) has two distinct parts (a,b^2) with a < b, and there are 6 ways to arrange these parts.

John Tyler Rascoe, Python program.

Cf. A000217, A000219, A000293, A164894, A379277.

Python
```
# see Links
```

a(n) = A379277(A164894(n)) for n > 0.

A385123 Triangle Read by rows: T(n,k) is the number of rooted ordered trees with n non-root nodes with non-root node labels in {1,..,k} such that all labels appear at least once in all groups of sibling nodes.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 5, 6, 6, 0, 14, 22, 36, 24, 0, 42, 90, 150, 240, 120, 0, 132, 378, 648, 1560, 1800, 720, 0, 429, 1638, 3318, 8400, 16800, 15120, 5040, 0, 1430, 7278, 18180, 43128, 126000, 191520, 141120, 40320, 0, 4862, 32946, 98502, 238320, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

John Tyler Rascoe, Jun 18 2025

			Triangle begins:
    k=0    1    2      3     4      5      6     7
 n=0 [1]
 n=1 [0,   1]
 n=2 [0,   2,   2]
 n=3 [0,   5,   6,     6]
 n=4 [0,  14,  22,    36,   24]
 n=5 [0,  42,  90,   150,  240,   120]
 n=6 [0, 132,  378,  648, 1560,  1800,   720]
 n=7 [0, 429, 1638, 3318, 8400, 16800, 15120, 5040]
...
T(3,2) = 6 counts the three leaf permutations of each of the following trees:
      __o__        __o__
     /  |  \      /  |  \
   (1) (1) (2)  (1) (2) (2)

Cf. A000108 (column k=1), A000142 (main diagonal), A385125 (row sums).

Cf. A107429, A384685, A384747.

subsets(S) = {my(s=List()); for(i=0, 2^(#S) -1, my(x=List()); for(j=1,#S, if(bitand(i, 1<<(j-1)), listput(x, S[j]))); listput(s,Vec(x))); Vec(s)}
C_aB(B) = {my(S = subsets(B)); sum(i=1,#S, (1/(1-x*z*#S[i]))*(-1)^(#B-#S[i]))}
D(k,N,B) = {if(k>N,1, substpol(C_aB(B),z,1 + D(k+1,N-#B+1,B)))}
Dx(N,B) = {Vec(1+D(1,N,B)+ O('x^(N+1)))}
T(max_row) = {my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Dx(N+1, vector(k,i,i))~])); vector(N, n, vector(n, k, v[n, k]))}
T(8)

cp's OEIS Frontend

User: John Tyler Rascoe

A386891 Irregular triangle read by rows: T(n,k) is the number of compositions of n such that the maximal cardinality of C is k, where C is a subset of the set of parts such that all elements in C appear in weakly increasing order within the composition.

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A385604 Number of compositions of n such that the odd parts are weakly increasing.

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A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

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Maple

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A386497 Number of sets of lists of [n] such that one list is the largest.

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A386375 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs more frequently than any other letter.

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Maple

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Formula

A386374 Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs at least as many times as any other letter.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

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Programs

Maple

Mathematica

PARI