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A347490
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Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 8.
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2
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1, 1, 9, 1, 73, 657, 1, 585, 4745, 42705, 384345, 1, 4681, 304265, 2738385, 22211345, 199902105, 1799118945, 1, 37449, 19477641, 156087945, 175298769, 11394419985, 92421406545, 102549779865, 831792658905, 7486133930145, 67375205371305, 1, 299593
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OFFSET
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1,3
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COMMENTS
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Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.
For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_8)^n with dimension increments (e_1,...,e_r).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.
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LINKS
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FORMULA
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T(n, (n)) = 1. T(n, L) = A022172(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.
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EXAMPLE
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The number of subspace chains 0 < V_1 < V_2 < (F_8)^3 is 657 = T(3, (1, 1, 1)). There are 73 = A022172(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 9 = A022172(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
k: 1 2 3 4 5
-----------------------
n=1: 1
n=2: 1 9
n=3: 1 73 657
n=4: 1 585 4745 42705 384345
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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